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# What is the first term of an arithmetic progression of positive intege

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What is the first term of an arithmetic progression of positive intege  [#permalink]

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25 Sep 2019, 04:34
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What is the first term of an arithmetic progression of positive integers ?

(1) Sum of the squares of the first and second term is 116.
(2) The seventh term is divisible by 10.

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Re: What is the first term of an arithmetic progression of positive intege  [#permalink]

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25 Sep 2019, 07:39
Bunuel wrote:
What is the first term of an arithmetic progression of positive integers ?

(1) Sum of the squares of the first and second term is 116.
(2) The seventh term is divisible by 10.

#1
Sum of the squares of the first and second term is 116.
only possiblity ; 4^2+10^2 ; 116
sufficient
#2
The seventh term is divisible by 10
insufficient
to determine as series progression not know
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Re: What is the first term of an arithmetic progression of positive intege  [#permalink]

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25 Sep 2019, 12:26
Bunuel wrote:
What is the first term of an arithmetic progression of positive integers ?

(1) Sum of the squares of the first and second term is 116.
(2) The seventh term is divisible by 10.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since an arithmetic progression has two variables, which are its first term and its difference and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Assume $$a_1$$ is its first term and $$a_2$$ is its second term.
If $$a_1 = 4$$ and $$a_2 = 10$$, then we have a difference $$6$$ and the seventh term $$a_7 = 4 + 6*6 = 40$$ which is divisible by $$10$$, $$a_1$$ is a positive integer and then answer is 'yes'.
If $$a_1 = -4$$ and $$a_2 = -10$$, then we have a difference $$-6$$ and the seventh term $$a_7 = -4 + 6*(-6) = -40$$ which is divisible by $$10$$, $$a_1$$ is a negative integer and then answer is 'no'.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: What is the first term of an arithmetic progression of positive intege  [#permalink]

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25 Sep 2019, 12:29
MathRevolution
the question stem says What is the first term of an arithmetic progression of positive integers

MathRevolution wrote:
Bunuel wrote:
What is the first term of an arithmetic progression of positive integers ?

(1) Sum of the squares of the first and second term is 116.
(2) The seventh term is divisible by 10.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since an arithmetic progression has two variables, which are its first term and its difference and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Assume $$a_1$$ is its first term and $$a_2$$ is its second term.
If $$a_1 = 4$$ and $$a_2 = 10$$, then we have a difference $$6$$ and the seventh term $$a_7 = 4 + 6*6 = 40$$ which is divisible by $$10$$, $$a_1$$ is a positive integer and then answer is 'yes'.
If $$a_1 = -4$$ and $$a_2 = -10$$, then we have a difference $$-6$$ and the seventh term $$a_7 = -4 + 6*(-6) = -40$$ which is divisible by $$10$$, $$a_1$$ is a negative integer and then answer is 'no'.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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What is the first term of an arithmetic progression of positive intege  [#permalink]

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07 Dec 2019, 05:25
1
Does a progression by definition has to be infinite?
Because if not, statement (1) leads to two possible progressions:

(4, 10, ....etc.)
or
(10, 4) and the progression just has 2 terms.

The first term could be 4 as well as 10. Therefore, statement (1) alone would not be sufficient and the answer would not be A. (but C instead)
Can someone pls help clarifying this?
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What is the first term of an arithmetic progression of positive intege  [#permalink]

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07 Dec 2019, 09:34
But in second series 10, 4...you will get negative values in finite progression. but question stem says that progression contain positive integers.

So all integers are positive.
Therefore, A is sufficient.

OhMy wrote:
Does a progression by definition has to be infinite?
Because if not, statement (1) leads to two possible progressions:

(4, 10, ....etc.)
or
(10, 4) and the progression just has 2 terms.

The first term could be 4 as well as 10. Therefore, statement (1) alone would not be sufficient and the answer would not be A. (but C instead)
Can someone pls help clarifying this?
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Status: Climbing the GMAT mountain
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What is the first term of an arithmetic progression of positive intege  [#permalink]

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08 Dec 2019, 01:54
gvij2017
Thanks for the response!

But when the progression just consists of two terms (10 and 4), then its finite and still only contains positive integers.

gvij2017 wrote:

But in second series 10, 4...you will get negative values in finite progression. but question stem says that progression contain positive integers.

So all integers are positive.
Therefore, A is sufficient.
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What is the first term of an arithmetic progression of positive intege  [#permalink]

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10 Dec 2019, 21:38
OhMy wrote:
gvij2017
Thanks for the response!

But when the progression just consists of two terms (10 and 4), then its finite and still only contains positive integers.

gvij2017 wrote:

But in second series 10, 4...you will get negative values in finite progression. but question stem says that progression contain positive integers.

So all integers are positive.
Therefore, A is sufficient.

Hi OhMy,
I think you are correct .

(1 ) Sum of the squares of the first and second term is 116.
4,10
10, 4
INSUFF.

(2) The seventh term is divisible by 10.
No idea of First term . INSUFF.

1+2

4 10 16 22 28 34 40
10 4 -2 -8 -14 -20 -26

Ans- Should be C
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What is the first term of an arithmetic progression of positive intege   [#permalink] 10 Dec 2019, 21:38
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