Multiple choice

Consider a sequence of real numbers b1, b2, ......., bn such that the sum of the first n terms is given by b1 + b2 + ....... + bn = 5n2 + 8n + 12, for all integers n ≥ 1. If br = 153, find the value of r.

  1. 10

  2. 12

  3. 15

  4. 18

  5. a
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C Correct answer
Explanation

The n-th term b_n is given by S_n - S_(n-1). Here, b_n = (5n^2 + 8n + 12) - (5(n-1)^2 + 8(n-1) + 12) = 10n + 3. Setting 10r + 3 = 153 gives 10r = 150, so r = 15.

AI explanation

To find an individual term of the sequence, subtract the sum of the first (n-1) terms from the sum of the first n terms. The sum of the first r terms is given as 5r^2 + 8r + 12, so the rth term br = (5r^2 + 8r + 12) minus (5(r-1)^2 + 8(r-1) + 12). Simplifying this expression yields br = 10r + 3, and setting this equal to 153 gives 10r = 150. Solving for r, we find that r is 15.

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