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Math Help - 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  -5,a_1,a_2, ... ,a_n,15 , and the sum of its first  n+2 terms is  100 , find  n.

    the answer is this:
     S=\frac{(n+2)(-5+15)}{2}=5n+10=100
    \therefore n=18

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

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

    15 is the (n+2)th term because  -5 is the first term, a_1 the second, ..., a_n is the (n+1)th term, hence 15 is the (n+2)th term.
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    Re: sum of arithmetic sequence

    Quote Originally Posted by muddywaters View Post
    Given the arithmetic sequence  -5,a_1,a_2, ... ,a_n,15 , and the sum of its first  n+2 terms is  100 , find  n.
    My question is this:
    the  \frac{(n+2)(-5+15)}{2} part is from  \frac{n(a+l)}{2}, where n is the number of terms, a is the first term and  l is the last term.
    In this case, for the sum of n+2 terms, the last term is the  (n+2)_t_hterm. the answer uses 15, which is the (n+1)_t_h term, since it's directly after the n_t_h term. it's not the  (n+2)_t_h term. why is it like that? did i misunderstand something??
    In the sequence  -5,a_1,a_2, ... ,a_n,15 there are (n+2) terms.
    The first term is -5.
    The second term is a_1.
    \;~\;~ \vdots
    The last term is 15.
    Thanks from muddywaters
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