If an ≠ 0 and n is a positive integer, is n odd?

[BTGmoderatorDC]

If an ≠ 0 and n is a positive integer, is n odd?

(1) a^n + a^(n + 1) < 0

(2) a is an integer.



OA C

Source: Manhattan Prep

[Jay@ManhattanReview]

If an ≠ 0 and n is a positive integer, is n odd?

(1) a^n + a^(n + 1) < 0

(2) a is an integer.

OA C

Source: Manhattan Prep

Given: an ≠ 0 and n is a positive integer;

Since an ≠ 0, we have a ≠ 0.

We have to ascertain whether is n odd.

Let's take each statement one by one.

(1) a^n + a^(n + 1) < 0

a^n + a^n*a < 0

a^n*(1 + a) < 0

=> the product of a^n and (1 + a) is negative. It is possible in two cases: one of them is positive and the other is negative.

Case 1: Say a^n < 0 and (1 + a) > 0

Given that 1 + a > 0, we have a > –1. Since a^n is negative, a must be negative and n must be odd. Thus, we have 0 < a < –1.

Case 2: Say a^n > 0 and (1 + a) < 0

Given that 1 + a < 0, we have a < –1. Since a^n is positive, and a is negative, we must have n even.

No unique value of n. Insufficient.

(2) a is an integer.

Certainly insufficient.

(1) and (2) together

In view of Statement 2, Case 1 discussed in Statement 1 is invalid; thus, as per Case 2, n is even. The answer is no: a unique answer. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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