Bunuel wrote:
For any given x and any positive integer n, the nth term in sequence S is defined by the equation \(S_n(x) = x^{(2n-1)}\). The product of all terms in sequence S from \(S_1(x)\) through \(S_k(x)\) for any positive integer k is equal to x raised to what power?
(A) k
(B) 2k – 1
(C) 2k
(D) (k^2+ 1)/2
(E) k^2
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:Start by listing simple terms of sequence S in order.
\(S_1(x) = x^{(2(1)-1)} = x^1 = x\)
\(S_2(x) = x^{(2(2)-1)} = x^3\)
\(S_3(x) = x^{(2(3)-1)} = x^5\)
By this point, we would hopefully notice that the exponents are the positive odd numbers in order.
Next, we need to consider the product of several of these terms, starting from \(S_1(x)\) on up to the kth term.
If k = 1, then the product of all the terms is simply x (since there is only the first term). The exponent here is 1. If we check the answer choices at this point, we could eliminate (C), because if we plug in k = 1, we get an incorrect result here (that is, 2). All the other answer choices give us 1, the correct result, so we have to keep going.
If k = 2, then the product of the first two terms is \(x*x^3 = x^4\). So we need to get 4 out if we plug in k = 2. Of the remaining answer choices, only (E) works. We can stop here.
If we really wanted to, we could check another value of k, or we could notice that we’re summing up the first k odd integers (because we’re multiplying the terms, which have the same base x and those integers as exponents). The sum of the first k odd integers equals k^2.
The correct answer is E.
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