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11 Oct 2021, 03:17
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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
(1) \(a_1=17\)
(2) The sum of the terms in the sequence is 77
11 Oct 2021, 03:17
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Bookmarks
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
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
