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Math Help - arithmetic sequence

  1. #1
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    arithmetic sequence

    hello
    'find the arithmetic progression of nine continuous numbers where the sum total is 135'
    kindly get me detailed solution
    sincerely
    anil
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  2. #2
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    Quote Originally Posted by anilpadavil View Post
    hello
    'find the arithmetic progression of nine continuous numbers where the sum total is 135'
    kindly get me detailed solution
    sincerely
    anil
    The first number is a the next is a+k the next is a+2k and so on ... the ninth is a+8k.

    In sum we have,
    \underbrace{(a+a+...+a)}_9 + k(1+2+...+8) = 9a+36k.

    So we want,
    9a+36k=135
    Divide by 9,
    a+4k=15.
    So the possible such sequences are:
    a=11,k=1
    a=7,k=2
    a=3,k=3
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  3. #3
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    Hello, Anil!

    Find the arithmetic progression of nine continuous numbers where the sum total is 135
    Did you mean "consecutive" numbers?
    . . Then the first term is a, and the common difference is d = 1.


    The sum of the first n terms of an A.P. is: . S_n\:=\:\frac{n}{2}\left[2a + (n-1)d\right]
    . . Then we have: . S_9 \:=\:\frac{9}{2}\left[2a + 8(1)\right] \:=\:135\quad\Rightarrow\quad a \:=\:11


    Therefore, the progression is: . 11,\,12,\,13,\,14,\,15,\,16,\,17,\,18,\,19

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