For any sequence question, it is usually a good idea to work out the first few terms until you understand the structure of the sequence. Here, if
s_n = x^(2n - 1)
then plugging in n =1, 2 and 3 we find:
s_1 = x^1
s_2 = x^3
s_3 = x^5
and so on. We want to multiply these terms together, which means we'll add the exponents. So the question is just asking: what is the sum of the odd numbers from 1 up to 2k - 1?
There are a couple of ways to finish the problem. We just worked out that, when k=3, the first three terms are x^1, x^3, and x^5, which have a product of x^9. So when k=3, the answer to the question is 9. Plugging k=3 into each answer choice, only E works, so E must be right (you need to try k=3 here, because if you try k=1 or k=2, you still have at least two candidate answer choices). Or we can do this algebraically. We want to add:
1, 3, 5, ..., 2k-1
This is an equally spaced list, increasing by 2 each time, so the mean of the list is equal to the average of the smallest and largest terms. So the mean of the list is (1 + 2k - 1)/2 = k. To find the number of terms n in any equally spaced list, we can use the formula:
n = (range / space) + 1
and here the range is (2k-1)-1 = 2k-2, and the spacing is 2, so
n = (2k-2)/2 + 1 = k-1+1 = k
So the mean of the list is k, the number of terms is k, and from the definition of the average, sum = avg*n, so sum = k*k = k^2.
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