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Duccio
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Bunuel's Algebra PS Diagnostic Test 14 Mar 2025, 02:27
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it is possible to have the explanation of the 22nd?
I don't understand why answering none of the above it's correct. In my opinion, the answer would be only third.
thank you.
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Bunuel
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Bunuel's Algebra PS Diagnostic Test 14 Mar 2025, 02:59
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Duccio
it is possible to have the explanation of the 22nd?
I don't understand why answering none of the above it's correct. In my opinion, the answer would be only third.
thank you.
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The solution is here: https://gmatclub.com/forum/bunuel-s-alg ... l#p3444574
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Bunuel's Algebra PS Diagnostic Test 16 Mar 2025, 19:51
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for statment 2 the only thing making it untrue is the zero case, otherwise it would be true.

Bunuel
Question 28 - Official Solution:


If \(45x = 121y\), which of the following must be true?

I. \(x > y\)

II. \(x^2 > y^2\)

III. \(\frac{x}{11}\) is an integer




A. I only
B. II only
C. III only
D. I and III only
E. None of the above


It is important to note that the problem does not specify that \(x\) and \(y\) are necessarily integers, nor does it require \(x\) and \(y\) to be necessarily positive. Therefore, when evaluating each option, it is essential to keep in mind that these variables may take on non-integer or non-positive values!

I. \(x > y\).

This statement is not always true, as \(x\) and \(y\) are not necessarily positive. For instance, consider \(x = -121\) and \(y=-45\), or \(x = 0\) and \(y=0\).

II. \(x^2 > y^2\).

This statement is not always true, as \(x\) and \(y\) are not necessarily integers. For example, consider \(x = \frac{1}{45}\) and \(y=\frac{1}{121}\). Additionally, since \(x\) and \(y\) can both be 0, this statement is also not true in that case.

III. \(\frac{x}{11}\) is an integer.

This statement is also not always true, as \(x\) and \(y\) are not necessarily integers. For instance, consider \(x = \frac{1}{45}\) and \(y=\frac{1}{121}\).


Answer: E
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Bunuel's Algebra PS Diagnostic Test 06 Apr 2025, 12:20
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Hi can you help me with clarity on the second step after squaring the equation? how did the first term change to (4 - \sqrt{15}) from ((\sqrt{4-\sqrt{15}})

Bunuel
Question 30 - Official Solution:

What is the value of \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) ?

A. \(-4\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)


We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.

\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)
Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:

\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)
Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::

\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)

\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)

\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)

\(=4(16-15)=4\).
Since the square of the expression is 4, the expression itself must be either 2 or -2. However, since the product of two positive terms and one negative term is negative (\((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})=positive*positive*negative=negative\)), we know that \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) is negative. Therefore, the final answer is -2.


Answer: B­
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Bunuel's Algebra PS Diagnostic Test 06 Apr 2025, 12:33
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Yug1812
Hi can you help me with clarity on the second step after squaring the equation? how did the first term change to (4 - \sqrt{15}) from ((\sqrt{4-\sqrt{15}})

Bunuel
Question 30 - Official Solution:

What is the value of \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) ?

A. \(-4\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)


We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.


\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:


\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::


\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)

\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)

\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)

\(=4(16-15)=4\).

Since the square of the expression is 4, the expression itself must be either 2 or -2. However, since the product of two positive terms and one negative term is negative (\((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})=positive*positive*negative=negative\)), we know that \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) is negative. Therefore, the final answer is -2.


Answer: B­
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Because the whole expression, \( (\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}) \) is squared and when squaring the first term, \( (\sqrt{4-\sqrt{15}})\), we loose the outer square root and get \( 4-\sqrt{15} \).

Hope it's clear.
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Bunuel's Algebra PS Diagnostic Test 03 Jul 2025, 06:35
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hi Bunuel, in this series please provide solution for q. no 14.
Bunuel
Question 30 - Official Solution:

What is the value of \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) ?

A. \(-4\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)


We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.


\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:


\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::


\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)

\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)

\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)

\(=4(16-15)=4\).


Answer: B­
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Bunuel's Algebra PS Diagnostic Test 03 Jul 2025, 06:44
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shaliny
hi Bunuel, in this series please provide solution for q. no 14.

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Check here: https://gmatclub.com/forum/bunuel-s-alg ... l#p3593656
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Bunuel's Algebra PS Diagnostic Test 06 Jul 2025, 23:31
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can't we get the answer by putting approx. values of square roots???
Bunuel
Question 30 - Official Solution:

What is the value of \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) ?

A. \(-4\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)


We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.


\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:


\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::


\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)

\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)

\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)

\(=4(16-15)=4\).


Answer: B­
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Bunuel's Algebra PS Diagnostic Test 06 Jul 2025, 23:35
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can't we get the answer by putting approx. values of square roots???
Bunuel
Question 30 - Official Solution:

What is the value of \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\) ?

A. \(-4\)
B. \(-2\)
C. \(-1\)
D. \(1\)
E. \(2\)


We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.


\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)

\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:


\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)

Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::


\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)

\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)

\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)

\(=4(16-15)=4\).


Answer: B­
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Yes, we can check here: https://gmatclub.com/forum/m26-184440.html
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Bunuel's Algebra PS Diagnostic Test 06 Jul 2025, 23:55
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hii Bunuel,
here is the confusion that in exponent when we write suppose (x ^2)^2 then we write it as x^4 so here for ((4^3)^2)^1, can't we write it as 4^6 ????
Bunuel
Question 23 - Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C
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Bunuel's Algebra PS Diagnostic Test 07 Jul 2025, 00:01
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hii Bunuel,
here is the confusion that in exponent when we write suppose (x ^2)^2 then we write it as x^4 so here for ((4^3)^2)^1, can't we write it as 4^6 ????
Bunuel
Question 23 - Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C
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Please check the highlighted part.

In (x^2)^2, yes, we use the rule (a^m)^n = a^(mn), so (4^3)^2 = 4^(3*2) = 4^6.

But in the original question, we don't have parentheses. We have 4^3^2, which, using a^m^n = a^(m^n), equals 4^(3^2), which is 4^9, not 4^6.
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Bunuel's Algebra PS Diagnostic Test 07 Jul 2025, 00:06
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means here is the difference just because of parentheses.. got you.. thanku
Bunuel
shaliny
hii Bunuel,
here is the confusion that in exponent when we write suppose (x ^2)^2 then we write it as x^4 so here for ((4^3)^2)^1, can't we write it as 4^6 ????
Bunuel
Question 23 - Official Solution:

What is the value of \(4^{3^{2^{1^{2^{3^4}}}}}\)?

A. \(2,144\)
B. \(262,142\)
C. \(262,144\)
D. \(262,146\)
E. \(262,148\)


To evaluate \(4^{3^{2^{1^{2^{3^4}}}}}\), we need to remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example, \(a^{m^n}\) means we first compute \(m^n\) and then use that result as the exponent for \(a\). Therefore, \(a^{m^n} = a^{(m^n)}\). On the other hand, if we have \((a^m)^n\), we compute \(a^m\) first and then raise the result to the power of \(n\), so \((a^m)^n=a^{mn}\).

Let's evaluate the exponents of 4 in our expression. Since \(1^{2^{3^4}}\) equals 1, we can ignore it completely, which means we will be left with \(4^{3^2}\).

Using the exponent rule, \(4^{3^2} = 4^9\).

Since the GMAT is a timed test, we may not have the luxury of actually computing the value of \(4^9\). Instead, we need to use some shortcuts to arrive at the answer. Fortunately, we can notice that the units digit of each answer choice is different, and if we can determine the units digit of \(4^9\), we can identify the correct answer.

The units digit of \(4^k\), where \(k\) is a positive integer, alternates between 4 and 6 for odd and even values of \(k\), respectively. Since 9 is an odd number, the units digit of \(4^9\) is 4.

We can now eliminate any answer choices that do not end in 4. The only answer choice that ends in 4 is C, so our answer is \(4^{3^{2^{1^{2^{3^4}}}}} = 262,144\).


Answer: C

Please check the highlighted part.

In (x^2)^2, yes, we use the rule (a^m)^n = a^(mn), so (4^3)^2 = 4^(3*2) = 4^6.

But in the original question, we don't have parentheses. We have 4^3^2, which, using a^m^n = a^(m^n), equals 4^(3^2), which is 4^9, not 4^6.
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Bunuel's Algebra PS Diagnostic Test 07 Jul 2025, 06:26
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hii please elaborate it i am not getting.
i am not getting that how the sequence be arranged. Is it wrong if i do as for d>0 - a1.....a16......a31.....a46 and for d<0 - a46....a31.......a16.......a1 or as you have written. what would be the right way?
how we decide 16th largest term and 31 smallest term in increasing or decreasing AP? quite confusing...
Bunuel
Question 22 - Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)

I. The 31st smallest term is zero.

II. The 16th largest term is zero.

III. The sum of the largest and smallest terms of the sequence is positive.

A. I only
B. II only
C. III only
D. I and II only
E. None of the above

Firstly, note that we don't know whether the sequence is increasing or decreasing.

If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)
As we can see, none of the options MUST be true.

Answer: E­
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Bunuel's Algebra PS Diagnostic Test Updated on: 07 Jul 2025, 07:40
Originally posted by Bunuel on 07 Jul 2025, 07:20.
Last edited by Bunuel on 07 Jul 2025, 07:40, edited 1 time in total.
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shaliny
hii please elaborate it i am not getting.
i am not getting that how the sequence be arranged. Is it wrong if i do as for d>0 - a1.....a16......a31.....a46 and for d<0 - a46....a31.......a16.......a1 or as you have written. what would be the right way?
how we decide 16th largest term and 31 smallest term in increasing or decreasing AP? quite confusing...
Bunuel
Question 22 - Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)

I. The 31st smallest term is zero.

II. The 16th largest term is zero.

III. The sum of the largest and smallest terms of the sequence is positive.

A. I only
B. II only
C. III only
D. I and II only
E. None of the above

Firstly, note that we don't know whether the sequence is increasing or decreasing.

If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)
As we can see, none of the options MUST be true.

Answer: E­
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A sequence is an ordered set of numbers. In our case, we have an arithmetic progression, so it's either increasing, decreasing, or constant if the common difference is zero.

  • If it's increasing (\(d > 0\)), we have \(a_1 < a_2 < ... < a_{46}\).
  • If it's decreasing (\(d < 0\)), we have \(a_1 > a_2 > ... > a_{46}\).

The terms stay in their original positions (\(a_1, a_2, ..., a_{46}\)), but whether a particular term is one of the largest or smallest depends on whether the progression is increasing or decreasing.
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Bunuel's Algebra PS Diagnostic Test 08 Jul 2025, 08:05
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means for 16th largest term we will see AP with d>0 and a16 = 16 th largest term
for 31st smallest term in AP with d<0 and a31st =31 smallest term, am i getting right or not??
Bunuel
shaliny
hii please elaborate it i am not getting.
i am not getting that how the sequence be arranged. Is it wrong if i do as for d>0 - a1.....a16......a31.....a46 and for d<0 - a46....a31.......a16.......a1 or as you have written. what would be the right way?
how we decide 16th largest term and 31 smallest term in increasing or decreasing AP? quite confusing...
Bunuel
Question 22 - Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)

I. The 31st smallest term is zero.

II. The 16th largest term is zero.

III. The sum of the largest and smallest terms of the sequence is positive.

A. I only
B. II only
C. III only
D. I and II only
E. None of the above

Firstly, note that we don't know whether the sequence is increasing or decreasing.

If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)
As we can see, none of the options MUST be true.

Answer: E­

A sequence is an ordered set of numbers. In our case, we have an arithmetic progression, so it's either increasing, decreasing, or constant if the common difference is zero.

  • If it's increasing (\(d > 0\)), we have \(a_1 < a_2 < ... < a_{46}\).
  • If it's decreasing (\(d < 0\)), we have \(a_1 > a_2 > ... > a_{46}\).

The terms stay in their original positions (\(a_1, a_2, ..., a_{46}\)), but whether a particular term is one of the largest or smallest depends on whether the progression is increasing or decreasing.
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Bunuel's Algebra PS Diagnostic Test 08 Jul 2025, 08:32
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shaliny
means for 16th largest term we will see AP with d>0 and a16 = 16 th largest term
for 31st smallest term in AP with d<0 and a31st =31 smallest term, am i getting right or not??

Bunuel
Question 22 - Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)

I. The 31st smallest term is zero.

II. The 16th largest term is zero.

III. The sum of the largest and smallest terms of the sequence is positive.

A. I only
B. II only
C. III only
D. I and II only
E. None of the above

Firstly, note that we don't know whether the sequence is increasing or decreasing.

If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)
As we can see, none of the options MUST be true.

Answer: E­
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The sequence is always written in the original order: \(a_1, a_2, ..., a_{46}\), no matter whether \(d > 0\) or \(d < 0\). We never reverse or rearrange the sequence itself.

  • In an increasing AP (\(d > 0\)), the 16th largest term is \(a_{31}\) and the 31st smallest term is \(a_{31}\).

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)

  • In a decreasing AP (\(d < 0\)), the 16th largest term is \(a_{16}\) and the 31st smallest term is \(a_{16}\).

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)

So the positions of the terms (\(a_{16}\), \(a_{31}\)) stay the same, but whether they count as the largest or smallest depends on the direction of the sequence.

It would help if you write out some example sequences, one increasing and one decreasing, to check this for yourself.
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Bunuel's Algebra PS Diagnostic Test 08 Jul 2025, 10:06
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Can someone explain why option III cannot be true
Bunuel
Question 22 - Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)

I. The 31st smallest term is zero.

II. The 16th largest term is zero.

III. The sum of the largest and smallest terms of the sequence is positive.

A. I only
B. II only
C. III only
D. I and II only
E. None of the above

Firstly, note that we don't know whether the sequence is increasing or decreasing.

If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46}\)
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:

\(a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46}\)
As we can see, none of the options MUST be true.

Answer: E­
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Bunuel's Algebra PS Diagnostic Test 10 Jul 2025, 14:37
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Shouldn't question 30 be either 2 or -2, not just -2?
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Bunuel's Algebra PS Diagnostic Test 10 Jul 2025, 15:20
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eshika23
Shouldn't question 30 be either 2 or -2, not just -2?
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\((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\)

No. The firs term, \( (\sqrt{4-\sqrt{15}}) \), is positive; the second term, \( (4 + \sqrt{15})\), is positive; and the third term, \( (\sqrt{6} - \sqrt{10}) \) is negative, so the whole product is negative.
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Bunuel's Algebra PS Diagnostic Test 28 Oct 2025, 15:26
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how to solve Question 3?
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