sum of arithmetic sequence

I'm sorry if this is in the wrong subforum, I am not sure where to put it. But I searched this topic and saw others posting in pre-calc so im posting this here.

Given the arithmetic sequence $\displaystyle -5,a_1,a_2, ... ,a_n,15 $, and the sum of its first $\displaystyle n+2 $ terms is $\displaystyle 100 $, find $\displaystyle n$.

the answer is this:

$\displaystyle S=\frac{(n+2)(-5+15)}{2}=5n+10=100$

$\displaystyle \therefore n=18$

My question is this:

the $\displaystyle \frac{(n+2)(-5+15)}{2} $ part is from $\displaystyle \frac{n(a+l)}{2}$, where $\displaystyle n$ is the number of terms, $\displaystyle a$ is the first term and $\displaystyle l$ is the last term.

In this case, for the sum of $\displaystyle n+2$ terms, the last term is the $\displaystyle (n+2)_t_h$term. the answer uses $\displaystyle 15$, which is the $\displaystyle (n+1)_t_h$ term, since it's directly after the $\displaystyle n_t_h$ term. it's not the $\displaystyle (n+2)_t_h$ term.

why is it like that? did i misunderstand something??

Re: sum of arithmetic sequence

$\displaystyle 15$ is the $\displaystyle (n+2)$th term because $\displaystyle -5$ is the first term, $\displaystyle a_1$ the second, ..., $\displaystyle a_n$ is the $\displaystyle (n+1)$th term, hence $\displaystyle 15$ is the $\displaystyle (n+2)$th term.

Re: sum of arithmetic sequence

Quote:

Originally Posted by

**muddywaters** Given the arithmetic sequence $\displaystyle -5,a_1,a_2, ... ,a_n,15 $, and the sum of its first $\displaystyle n+2 $ terms is $\displaystyle 100 $, find $\displaystyle n$.

My question is this:

the $\displaystyle \frac{(n+2)(-5+15)}{2} $ part is from $\displaystyle \frac{n(a+l)}{2}$, where $\displaystyle n$ is the number of terms, $\displaystyle a$ is the first term and $\displaystyle l$ is the last term.

In this case, for the sum of $\displaystyle n+2$ terms, the last term is the $\displaystyle (n+2)_t_h$term. the answer uses $\displaystyle 15$, which is the $\displaystyle (n+1)_t_h$ term, since it's directly after the $\displaystyle n_t_h$ term. it's not the $\displaystyle (n+2)_t_h$ term. why is it like that? did i misunderstand something??

In the sequence $\displaystyle -5,a_1,a_2, ... ,a_n,15 $ there are $\displaystyle (n+2)$ terms.

The first term is $\displaystyle -5$.

The second term is $\displaystyle a_1$.

$\displaystyle \;~\;~ \vdots $

The last term is $\displaystyle 15$.