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The sum of the first k terms of a series S is 3k^2 + 5k . Which one of the following is correct?

Options

  1. AThe terms of S form an arithmetic progression with common difference 14.
  2. BThe terms of S form an arithmetic progression with common difference 6.
  3. CThe terms of S form a geometric progression with common ratio 10 7 .
  4. DThe terms of S form a geometric progression with common ratio 11 4 .

Correct answer

B. The terms of S form an arithmetic progression with common difference 6.

Step-by-step solution

The sum of the first k terms of the series is given by S_k = 3k^2 + 5k . The k -th term of the series, T_k , is obtained by subtracting the sum of the first k-1 terms from the sum of the first k terms: T_k = S_k - S_ k-1 Substituting the given expression for S_k and S_ k-1 : T_k = (3k^2 + 5k) - [3(k-1)^2 + 5(k-1)] T_k = 3k^2 + 5k - [3(k^2 - 2k + 1) + 5k - 5] T_k = 3k^2 + 5k - (3k^2 - 6k + 3 + 5k - 5) T_k = 3k^2 + 5k - (3k^2 - k - 2) T_k = 6k + 2 To find the common difference, we calculate the difference between con

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