Write the sum without using sigma notation. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$
A) $2 ^ { 1 } + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { 10 }$
B) $1 \cdot 4 ^ { 1 } + 2 \cdot 4 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 4 ^ { 10 }$
C) $1 \cdot 2 ^ { 1 } + 2 \cdot 2 ^ { 2 } + 3 \cdot 2 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$
D) $1 \cdot 2 ^ { 1 } + 2 \cdot 3 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$
E) none of these

Question

Quizplus · Accepted Answer

The sum represents the summation of the first 10 terms of a geometric series with common ratio 2. The formula for the sum of the first n terms of a geometric series with first term a and common ratio r is 
$$S_n = a\frac{1-r^n}{1-r}$$
Plugging in a = 2, r = 2, and n = 10, we get 
$$S_{10} = 2\frac{1-2^{10}}{1-2} = 2\cdot(2^{10}-1) = 2^{11}-2$$
which is equivalent to the sum given in choice C.

Write the Sum Without Using Sigma Notation ∑n=110n⋅2n\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }∑n=110​n⋅2n

Write the Sum Without Using Sigma Notation $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$