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Math Help - Arithmetic progression

  1. #1
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    Arithmetic progression

    the  N^{th} term of an A.P is denoted by U_{n} and the sum of the 1^{st}n terms by S_{N} in a certain A.P ;; U_{5}+U_{16}=44 and  S_{18}=3S_{10}. calculate the value of the first term and the common difference.

    i have been trying to get two equations from the given equation using the rules for arithmetic progression but i have been unable to do so..

    i really need some help coming up with these equations.
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  2. #2
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    Quote Originally Posted by sigma1 View Post
    the  N^{th} term of an A.P is denoted by U_{n} and the sum of the 1^{st}n terms by S_{N} in a certain A.P ;; U_{5}+U_{16}=44 and  S_{18}=3S_{10}. calculate the value of the first term and the common difference.

    i have been trying to get two equations from the given equation using the rules for arithmetic progression but i have been unable to do so..

    i really need some help coming up with these equations.


    Using the formula or the n-th term in an A.P., for all n\,,\,\,u_n=u_1+(n-1)d\Longrightarrow 44=u_5+u_{16}= u_1+4d+u_1+15d=2u_1+19d\Longrightarrow (I)\,\,2u_1+19d=44 , and using the formula for the sum of consecutive elements in an A.P. we get

    3S_{10}=3\cdot \frac{10}{2}\left(2u_1+9d\right)=\frac{18}{2}\left  (2u_1+17d\right)=S_{18} \Longrightarrow 10u_1+45d=6u_1+51d\Longrightarrow (II)\,\,2u_1-3d=0 , and there you have two linear eq's in two unknowns: u_1\,\,\,and\,\,\,d

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