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  • An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Find the AP.

An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Find the AP.

Answers (1)

Total terms = 37,  middle term =

\frac{37+1}{2}=19 \\

Three middle most terms, 18th , 19thand 20th, given

 a\textsubscript{18} + a\textsubscript{19} + a\textsubscript{20} = 225 \\

\\a + 17d + a + 18d + a + 19d = 225 $ ( Using a\textsubscript{n} $= a + (n - 1)d ) \\ 3a + 54d = 225\\ $dividing by 3 we get\\$ a + 18d = 75\\ a = 75 - 18d $ \ldots $ (1)\\ $According to question\\$ a\textsubscript{35} + a\textsubscript{36} + a\textsubscript{37} = 429 ( $Using a\textsubscript{n} $= a + (n - 1)d ) \\ a + 34d + a+ 35d + a + 36d = 429\\ 3a + 105d = 429\\ $Dividing by three we get\\$ a + 35d = 143\\ $Put value from equation (1) we get\\$ 75 - 18d + 35d = 143, \\ 17d = 143 - 75\\ d = 68, \\ \begin{aligned} & d=\frac{68}{17} \ & d=4 \ \end{aligned} \\ $Put d = 4 in (1) we get\\$ a = 75 - 72 = 3\\ a + d = 3 + 4 = 7\\ a + 2d = 3 + 2(4) = 11\\ $Required AP is$ 3, 7, 11, $ \ldots $ $ \ldots $ .\\

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