GMAT Club World Cup 2022 (DAY 5): The infinite decreasing sequence

Statement 1 Only -
\(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\)

both \(a_{14}\) and \(a_{13}\) are either positive or negative. Hence, the last 12 numbers could be +ve or -ve.
INSUFFICIENT

Statement 2 Only -
The product of any two of the first 14 terms is positive.

All 14 terms could be +ve or -ve.
INSUFFICIENT

1 & 2 Together -
All 14 terms could be +ve or -ve.

INSUFFICIENT

Hence, the answer is E.

given that

\(a_{n} < a_{n-1}\), for all n > 1

#1
\(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).
which is 1
so a15 would be either 0 or - ve
we can say that all terms from 1 to 14 are all +ve
therefore sufficient to target

#2
(2) The product of any two of the first 14 terms is positive.
possible when all terms are either -ve or all are +ve
which cannot be determine as its an even count
insufficient
option A is correct

st.1

a14/a13 = sqrt(10)/pi

or a14 = (3.16/3.14) a13

but we are also told that a1>a2>a3-----

this could be true if both a14, and a13 are negative. if both are negative that means all previous terms must be negative, and also all the terms after a14 must also be negative. none of the terms in the sequence are positive.

sufficient

st. 2

sum of first fourteen terms is positive, which is possible if all of them are either negative or positive. since two different scenarios will result in two different outcome hence answer b alone is not sufficient

Correct answer A

(1) a14/a13=√10/π
if a13 and a14 are positive then yes
but if negative let's say -3 and -4
then LHS>RHS
both must be positive and there are more than 12 +ve numbers
sufficient

(2) The product of any two of the first 14 terms is positive.
can be -1 -2 -3 -4 .....
product of two positive but no positive numbers

can be 55 54 53 52...
insufficient

IMO A
product positive ,numbers positive

Using Statement 1,
Rt(10) = 3.16
and pi = 3.14

and since the successive terms have to be smaller, i.e., a14 < a13, we can conclude that both a14 and a13 are negative.
But does that help us solve the question? No.

Using statement 2,
We can conclude that either all of the 14 numbers are positive or all of the 14 numbers are negative. But, the golden question: does that help us answer the question? No.

Using both statements,
We know that the 13th and 14th terms are negatives, and it's either all of them are positive or all of them are negative, then in that case, the first 14 terms of the sequence have to be negatives.

Does this help us answer our question? Yes, it does.

Option C is the right answer.

Correct answer: Choice E

Intutitvely speaking :

Statment 1 tells nothing about the sign since it is division. numbers can be negative or positive

Statment 2 tells nothing about the sign since it is division. numbers can be negative or positive

Togther , the statments dont give any new info.

hence choice E

IMO-c


(1) a14/a13=root10/π
root 10 is equal to 3.16 and π is equal to 3.14
it is given in the question
a14<a13
means a14 and a 13 has negative sign but we dont know how many terms have negative sign.


(2) The product of any two of the first 14 terms is positive.
means till 14 all term have same sign

Using Both using both we can conclude till 14 all term have Negative sign.

From the given details in the question we know that

\(x_1 > x_2.....> x_{12} > x_{13} > x_{14} > ......\)

Statement 1

\(\frac{x_{14}}{x_{13}}\) = \(\frac{\sqrt{10}}{\pi}\)

As \(\sqrt{10}\) > \(\pi\), there was some factor that got cancelled between the numerator and denominator. If this were not the case, then \(x_{14}\) would have been greater than \(x_{13}\).

Also, had it been any positive factor \(x_{14} > x_{13} \), but we know from the question stem that \(x_{14} < x_{13} \), so the factor was negative.

With the above analysis, we can conclude that both \(x_{14}\) and \( x_{13} \) were negative.

Even if the series had positive integers, the maximum number integers that the series could have had is till \(x_{12}\)

Does the sequence have more than 12 positive numbers ? - No

Hence A is sufficient.

Statement 2

Case 1

Let's assume all the numbers in the series are positive numbers.

Does the sequence have more than 12 positive numbers ? - Yes

Case 2

Let's assume all the numbers in the series are negative numbers.

Does the sequence have more than 12 positive numbers ? - No

Hence this statement is not sufficient.

IMO A

1 is not sufficient, we dont know a13/a12

2 is not sufficient

from both statements we cant find a13/a12
so, answer E

Given \(a_{n} < a_{n-1}\), for all n > 1

Statement 1:
"\(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\)."

Now \(\frac{\sqrt{10}}{\pi}\) > 1
Therefore only way this can be satisfied in the given sequence is when a13 and a14 are both negative.
Now we can consider 2 cases
Case-1
a1,a2,..a12 all positives (decreasing in value) followed by a13,a14 which are negative and then followed by more negative numbers.
e.g. 20,19,18,17...4(=a12),-root(10)(=a13),-PI(=a14)
Case-2
a1,a2,a3.. a12 all negatives (decreasing in value) followed by a13 and a14 followed by more negatives.
e.g. -.5,-1,-2,-3.. -9(=a12),-3root(10),-3PI
SO in no way can the sequence have more than12 positive numbers.
SUFFICIENT


Statement 2:
"The product of any two of the first 14 terms is positive."

This can be possible in two possible ways:
Case 1- 30,25,24,23,22,21,20,19,18,17,16,15(=a12),14(=a13),12(=a14) ...
So in this case there are more that 12 positive numbers.
Case -2 (All negative numbers like ) -1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14 ...
So in this case there are no positive numbers.
INSUFFICIENT

Ans A

[quote="Bunuel"]The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_{n} < a_{n-1}\), for all n > 1. Does the sequence have more than 12 positive numbers ?


(1) \(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).

(2) The product of any two of the first 14 terms is positive.


Solution image Attached.A is the answer

Answer: C

The infinite sequence a1, a2,…, an, … is such that an<a(n−1), for all n > 1. Does the sequence have more than 12 positive numbers ?


(1) a14 / a13 = √10 /π.
=> a14 > a13
Since we already know that for the sequence, an < a(n-1)
therefore, both a14 and a13 needs to be negative.
so, any number following a14 in the sequence would be negative.

Now, for any number prior to a13 in the sequence i.e. a1, a2,....a13
These will be greater than a13 on number line. and can be all negative, all positive or combination of negative and positive.
So, it can't be deduced if more than 12 numbers will be positive.
Insufficient.

(2) The product of any two of the first 14 terms is positive.
Case 1: all first 14 terms are positive
then 12 numbers are positive in the sequence.
Case 2: all first 14 terms are negative.
then further numbers will also be negative as the sequence is a decreasing one.
=> No numbers will be positive.
We have two results.
Hence, Insufficient.

Both statements together.
a13 and a14 are negative.
For the product of any two of the first 14 terms to be positive, all first 12 terms need to be negative.
therefore first 14 terms are negative.
Now, the series is decreasing one. any term beyond 14th term will be less i.e. all negative.
=> There will be no positive number in the series.
sequence does not have more than 12 positive numbers.

Sufficient.

IMO Option A is correct.
Statement I shows that both a14 and a13 are both negatives. √10 is approximately 3.16 which is greater than π which is approximately 3.14.
Statement I is sufficient . Statement I tells me that the sequence does not have more than 12 positive numbers. The sequence members after the 14th term will all be negatives.

Statement II Not Sufficient. Statement II only tells me the first 14 terms are either all negatives or they are all positives.

Imo C

As per the definition, the series is an< an-1, which means
for n=2, a2<a1
for n=3, a3<a2 and so on...

Statement 1 a14/a13=√10/π = 3.16/3.14 >1 => Not possible if both the numbers are positive
It means both the number are negative to make a14/a13<1
But it doesn't give any information about the previous terms. They can be all negative or all positive.
Insufficient
Statement 2 The product of any two of the first 14 terms is positive.
It means either all the terms are negative or all are positive.
Insufficient

Combining both the statements
As we know from statement 1 that the 13th and 14th terms are negative
and hence from statement 2 all the terms are negative.
It means all the rest of terms in the series are also negative to make the an< an-1
Hence we have the definite 'No'
Sufficient

Given: An<An-1
Check : if sequence has more than 12 positive numbers

st1. A14/A13 = sqrt10/pi > 1.006
Implies Mod of A14 is greater than mod of A13, hence the numbers must be negative for condition to be true

since 13th term is negative hence, we cant have more than 12 positive numbers in the sequence. Hence sufficient

st2. product of any two numbers upto 14th term is positive it implies all numbers upto 14th term are negative and since the 15th term and beyond are all negative as they have to be less than the previous number
therefore sufficient to say that we cant have more than 12 positive numbers in the sequence

Answer : D

1) Sufficient
2) Insufficient

Ans - A

Posted from my mobile device

Given: The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_{n} < a_{n-1}\), for all n > 1.
Asked: Does the sequence have more than 12 positive numbers ?

Since it is infinite sequence such that \(a_{n} < a_{n-1}\), for all n>1, but common ratio is not provided.

(1) \(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).
\(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).
\(\frac{a_{14}}{a_{13}}=\frac{3.16}{3.14} = 1.006 > 1\).
Since \(a_14 < a_13\), both are negative
All other terms \(a_15\) onwards will be negative since \(a_{n} < a_{n-1}\)
Since common ratio is not provided, first 12 terms can be positive or negative.
NOT SUFFICIENT

(2) The product of any two of the first 14 terms is positive.
Either all frist 14 terms are positive or all first 14 terms are negative.
But signs of first 14 terms is unknown
NOT SUFFICIENT

(1) + (2)
(1) \(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).
\(\frac{a_{14}}{a_{13}}=\frac{\sqrt{10}}{\pi}\).
\(\frac{a_{14}}{a_{13}}=\frac{3.16}{3.14} = 1.006\).
Since \(a_{14} < a_{13}\) both are negative
All other terms \(a_{15}\) onwards will be negative since \(a_{n} < a_{n-1}\)
(2) The product of any two of the first 14 terms is positive.
Since \(a_{14} < a_{13}\) both are negative, all first 14 terms are negative.
All other terms \(a_{15}\) onwards will be negative since \(a_{n} < a_{n-1}\)
None of the terms of the sequence is positive
SUFFICIENT

IMO C