Multiple choice

Sum of the series to $n$ terms $5+7+13+31+85+\cdots $ is

  1. ${ 3 }^{ n }+8n+1$
  2. $\dfrac { 1 }{ 2 } \left[ { 3 }^{ n }+8n-1 \right] $
  3. $\dfrac { 1 }{ 2 } \left( { 3 }^{ n }+8n+1 \right) $

  4. None of the above
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B Correct answer
AI explanation

The differences between consecutive terms are 2, 6, 18, 27, which form a geometric progression with the first term 2 and common ratio 3. The nth term of the series can be expressed as the sum of this geometric progression plus an initial constant, yielding a_n = 3^(n-1) + 4. To find the sum of n terms, add the sum of the geometric progression, (3^n - 1)/2, to the sum of n ones multiplied by 4. This evaluates to 1/2[3^n + 8n - 1].

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