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Answer» $\small\bold{\underline{\sf{\blue{\:\:Given\;\::-}}}}$ SUM of the 8th Terms is 136 and that of first 15 terms is 465. To Find:- Sum of first 25 Terms = ? EXPLANATION:- We know that, $\bigstar\;\:\small{\underline{\boxed{\sf{\red{\dfrac{n}{<klux>2</klux>} 2a \:+\:(n\:-\:1)d}}}}}$ Here, a = First Term D = Common Difference $\bold{\underline{\sf{According\:to\:Ques\: Now}}}$ $\longrightarrow\sf\: \dfrac{8}{2} (2a \:+\:(8\:-\:1)d \:=\: 136$ $\longrightarrow\sf\: 2a + 7d = 136$ -------Eq(1) $\rule{150}2$ $\longrightarrow\sf\: \dfrac{15}{2} (2a \:+\:(15\:-\:1)d = 465$ $\longrightarrow\sf\: 2a \:+\:14d \:=\:\dfrac{465\times\: 2}{15}$ $\longrightarrow\sf\: 2a + 14d = 62$ -------Eq(2) $\rule{150}2$ $\dag\:\small\bold{\underline{\sf{\blue{Now,\: Subtracting\: Equation\:(1)\:from\:(2)}}}}$ $\longrightarrow\sf\: 2a \: + \: 17d \: =\: 136$ $\longrightarrow\sf\:2a \:+\: 14d \:=\: 62$ $\longrightarrow\sf\: - 7d = - 28$ $\longrightarrow\sf\: d = \dfrac{-28}{-7}$ $\longrightarrow\large\boxed{\sf{\red{d\:=\: 4}}}$ ★ Substituting the VALUE of d in Equation (1) $\longrightarrow\sf\: 2a + 7d = 34$ $\longrightarrow\sf\:2a + 7(4) = 34$ $\longrightarrow\sf\:2a + 28 = 34$ $\longrightarrow\sf\:2a = 34 - 28$ $\longrightarrow\sf\:2a = 6$ $\longrightarrow\sf\:a = \cancel\dfrac{6}{2}$ $\longrightarrow\large{\sf{\red{a \:=\: 3}}}$ $\rule{150}2$ Now, Finding the Sum of 25 Terms $\longrightarrow\sf\: Sn = \dfrac{n}{2}(2a \:+\;(n \:-\:1)d$ $\longrightarrow\sf\: = \dfrac{25}{2}(6 + 24 \times \: 4)$ $\longrightarrow\sf\: \dfrac{25}{2} \times\: 102$ $\longrightarrow\large{\underline{\boxed{\sf{\pink{1275}}}}}$ $\small\bold{\underline{\sf{\blue{Hence,\;Sum\:of\: First\:25\;terms\: is\:1275}}}}$

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