Question

find the sum of the following series.
14 E n=1 (2n+1)
what is S10 for 1+3+9+27+...?

Answer 1

you can break up the sum into 2 different sums, like this:\[\sum_{n=1}^{14}\left( 2n+1 \right)=\sum_{n=1}^{14}\left( 2n \right)+\sum_{n=1}^{14}\left( 1 \right)\]You can then further refine things:
First, you can pull out any factors not involving the index (n)
\[\Rightarrow \sum_{n=1}^{14}\left( 2n \right)=2\sum_{n=1}^{14}\left( n \right)\]

Also, the sum of constant not involving the index is just the number of terms times the constant:
\[\sum_{n=1}^{14}\left( 1 \right)= \underbrace{1+1+\cdots +1}_\text{14 terms} = 14\cdot(1)=14\]

The multiplier here is 14 because we are summing 14 terms of 1.

Answer 2

Do you know the formula for the sum of integers?