What is the sum of the first 100 terms of this sequence: 5, 11, 17, 23...?
I know how to calculate this but according to a site, my calculations are wrong. I did 5 + 6*99 which indicates the final term. Adding the final term to the first term 599+5 and dividing that by 2, would be the average. The number of terms as stated is 100, so the answer should be 302 * 100, but according to the site, that's not the answer, so I don't know what I did wrong here.
The sum of an arithmetic sequence is given by $\displaystyle S_n = \dfrac{n}{2}(2a + (n-1)d)$ where
- $\displaystyle n$ is the number of terms
- $\displaystyle a = U_1$ which is the first term
- $\displaystyle d$ is the common difference (between terms)
You are told that $\displaystyle n=100$ and it's not hard to see that $\displaystyle a=5$. Now all that's left is to find the common difference and then plug into the formula at the top of this post.
L'edit: Google Calculator gives 30,200 as a answer which is evidently the same as 302*100
$\displaystyle 5+[5+6]+[5+(2)6]+[5+(3)6]+....+[5+(99)6]$
$\displaystyle =(100)5+6(1+2+3+....+99)$
$\displaystyle =500+6[(50-49)+(50-48)+.....+50+....+(50+48)+(50+49)]$
$\displaystyle =500+6(99)50$
However, you are probably expected to use the general formula for summing an arithmetic series,
having recognised the series as "arithmetic"
from the constant difference between terms.