GMAT CLUB OLYMPICS: The sequence of n integers is such that

Question

The sequence of  n  integers is such that  a_{k+1}=a_k-2  for k > 1. If  a_1  is a prime number, is n a prime number?

(1)  a_1=17 
(2) The sum of the terms in the sequence is 77

M36-23

Bunuel · Accepted Answer

Official Solution:

The sequence of  n  integers is such that  a_{k+1}=a_k-2  for  k > 1 . If  a_1  is a prime number, is  n  a prime number? 
 
(1)  a_1=17 
 
The sequence is: 17, 15, 13, 11, ... But we don't know the value of  n  (how long the sequence continues). Not sufficient.
 
(2) The sum of the terms in the sequence is 77
 
The sequence at hand is an arithmetic progression with the common difference of -2. The sum of the  n  terms of an arithmetic progression with the common difference of  d  is  \frac{n}{2}*(2a_1 + (n – 1)d)) .
 
So, we are told that  \frac{n}{2}*(2a_1 + (n – 1)(-2)))=77 ;
 
 n*(a_1 - n + 1))=77 
 
77 can be expressed as the product of two positive integers ( n  and  (a_1 - n + 1)  in our case) in the following 4 ways: 1*77, 7*11, 11*7 and 77*1.  n  cannot be 1 or 77 because in this case  a_1  is NOT a prime number (as we are given in the stem), so  n  is 7 or 11. Both are primes. Sufficient.

Answer: B

sumitkrocks · Answer

Answer is (B)

The sequence of  n  integers is such that  a_{k+1}=a_k-2  for k > 1. If  a_1  is a prime number, is n a prime number?

Statement (1)  a_1=17 
No information about n 
Not sufficient

Statement (2) The sum of the terms in the sequence is 77
a1 + a2 + a3 + a4...... =77
a1 + a1-2 + a1-4.....a1-2(n-1)
n*a1 -  \frac{2(n)(n-1)}{2}  = 77

n(a1+1-n) = 7*11 or 1*77

n=7 ; a1 =17
n=11; a1 =17
etc
In either case n is prime

N can be prime

(B) is the answer

Kinshook · Answer

Given: The sequence of  n  integers is such that  a_{k+1}=a_k-2  for k > 1. 
Asked: If  a_1  is a prime number, is n a prime number?

(1)  a_1=17 
 a_2 = 15 
 a_3 = 13 
 a_n = 17 - 2(n-1) = 19 - 2n 
Since a_n or n is unknown
NOT SUFFICIENT

(2) The sum of the terms in the sequence is 77
 a_n = a_1 - 2(n-1) = a_1+2 - 2n 
S_n = n/2 (2a_1 - 2(n-1)) = n(a_1+1-n) = 77
If  n = 1; a_1 + 1 - 1 = 77; a_1 = 77 ; Not feasible since  a_1  is a prime number
If  n = 7; 7(a_1 + 1-7) = 77; a_1 = 11 + 6 = 17 
If  n = 11; 11(a_1 + 1 - 11) = 77; a_1 = 10 +7 = 17 
In both the case when n =7 or n = 11, n is a prime number.
SUFFICIENT

IMO B

azeempanjwani14 · Answer

The question stem tells us that a1 is a prime number. We need to find if 'n' i.e. the number of integers is prime or not

Statement 1: 
It tells us that a1 = 17 
The sequence will be 17, 15, 13, 11, ...

We do not know the sum of these numbers or the number of elements in this sequence, 'n' can be prime # or a composite #, there are infinite possibilities; hence we cannot calculate the value of 'n'

Statement 1 is insufficient

Statement 2:  
It tells us that the sum of the terms in the sequence is 77

According to the information given in the statement, we can see that the first term is a1 and d (difference between 2 consecutive terms will be -2) therefore -

First term (a) = a1 
Difference (d) = -2 
Number of elements (n) = n

Sum of an AP sequence = (n/2)   
77 = (n/2)   ----- putting the value of d and sum 
77 = n (a - n - 1) ---- taking out 2 common in the RHS 
7 x 11 = n (a - n - 1) ---- factoring 77 = 7 x 11

From the above equation we can see that the two elements in the RHS can be: 
(1) n = 7; (a - n - 1) = 11 
or, 
(2) n = 11; (a - n - 1) = 7

in both the cases we can see that n = prime # (either 7 or 11)

Hence, second statement is sufficient (Option B is the correct answer)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

GMAT CLUB OLYMPICS: The sequence of n integers is such that

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

GMAT CLUB OLYMPICS: The sequence of n integers is such that

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards