Question

The sum of the series \( (1 + 2) + (1 + 2 + 2^2) + (1 + 2 + 2^2 + 2^3) + \cdots \) up to terms is:

• A. \( 2n^2 + n - 4 \)
• B. \( 2^n - 1 \)
• C. \( 2n^2 + n - 1 \)
• D. \( 2n^n - 1 \)

Hint:

When summing terms of a geometric series, use the formula for the sum of the first \( n \) terms to simplify the expression.

Answer 1

The Correct Option is A

We are given the series: \[ (1 + 2) + (1 + 2 + 2^2) + (1 + 2 + 2^2 + 2^3) + \cdots \] Step 1: Simplify the sum The general term of the series is: \[ S_n = (1 + 2 + 2^2 + 2^3 + \cdots + 2^n) \] This is a geometric series with the first term \( 1 \), common ratio \( 2 \), and \( n \) terms. The sum of the first \( n \) terms of a geometric series is: \[ S_n = \frac{2^{n+1} - 1}{2 - 1} = 2^{n+1} - 1 \] Thus, the sum of the first \( n \) terms is: \[ 2n^2 + n - 4 \] Thus, the correct answer is \( 2n^2 + n - 4 \).