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Thread: sum of arithmetic sequence

  1. #1
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    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??
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  2. #2
    MHF Contributor Siron's Avatar
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    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.
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    Re: sum of arithmetic sequence

    Quote Originally Posted by muddywaters View Post
    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 $\displaystyle 15$.
    Thanks from muddywaters
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