Question

What is the number of terms in the series 5, 7, 9,...if its sum is 1020 ?
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Answer 1

Let 1020 be the sum of first $\displaystyle\sf{n}$ terms of the sequence 5, 7, 9,... where $\displaystyle\sf{n\geq1,}$ i.e.,

$\displaystyle\longrightarrow\sf{S_n=1020}$

$\displaystyle\longrightarrow\sf{\dfrac{n}{2}\Big[2\times5+(n-1)(7-5)\Big]=1020}$

$\displaystyle\longrightarrow\sf{n\Big[10+2(n-1)\Big]=2040}$

$\displaystyle\longrightarrow\sf{n\Big[2n+8\Big]=2040}$

$\displaystyle\longrightarrow\sf{n(n+4)=1020}$

$\displaystyle\longrightarrow\sf{n^2+4n-1020=0}$

$\displaystyle\longrightarrow\sf{n^2+34n-30n-1020=0}$

$\displaystyle\longrightarrow\sf{(n-30)(n+34)=0}$

$\displaystyle\Longrightarrow\sf{\underline{\underline{n=30}}}$