Multiple choice

Find the sum to $n$ terms of the sequence $$ given by $a_{n} = 2^{n} + 3n, n\epsilon N$.

  1. $2(2^{n} + 1) + \dfrac {3n}{2}(n + 1)$
  2. $2(2^{n} - 1) + \dfrac {3n}{2}(n + 1)$
  3. $(2^{n} - 1) + \dfrac {3n}{2}(n + 1)$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sum is the sum of a geometric series (2^n) and an arithmetic series (3n). Sum(2^k) from 1 to n is 2(2^n - 1). Sum(3k) from 1 to n is 3n(n+1)/2. Adding these gives 2(2^n - 1) + 3n(n+1)/2.

AI explanation

Split the sequence into two separate series by writing the sum as the sum of 2^n plus the sum of 3n. The first part is a geometric progression with the first term 2 and common ratio 2, summing to 2(2^n - 1). The second part is an arithmetic progression with the first term 3 and common difference 3, summing to 3n/2 multiplied by (n + 1). Adding these two results gives 2(2^n - 1) + (3n/2)(n + 1).

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