Multiple choice

Find the third term of the sequences for which $S_n$ is given below. $S_n\,=\,n^{2}\,(n\,+\,1)$

  1. $32$
  2. $24$
  3. $52$
  4. $64$
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B Correct answer
Explanation

The n-th term a_n is given by S_n - S_(n-1). For n=3, a_3 = S_3 - S_2. S_3 = 3^2 * (3+1) = 9 * 4 = 36. S_2 = 2^2 * (2+1) = 4 * 3 = 12. Thus, a_3 = 36 - 12 = 24.

AI explanation

To find the third term, calculate the sum of the first three terms and subtract the sum of the first two terms. Using the given sum formula, S3 equals 3 squared times 4, which is 36, and S2 equals 2 squared times 3, which is 12. The difference, 36 minus 12, gives the third term as 24.

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