If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289,find the sum of first n terms

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289

1.	If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289,find the sum of first n terms
Answer» Let a be the first term and d be the common difference of given AP.Then, we haveS7=49{tex}\\Rightarrow \\frac { 7 } { 2 } [ 2 a + 6 d ] = 49{/tex}{tex}\\Rightarrow \\frac { 7 \\times 2 } { 2 } [ a + 3 d ] = 49{/tex}{tex}\\Rightarrow{/tex}a+3d=7...(i)Also, S17=289{tex}\\Rightarrow \\frac { 17 } { 2 } [ 2 a + 16 d ] = 289{/tex}{tex}\\Rightarrow \\frac { 17 \\times 2 } { 2 } [ a + 8 d ] = 289{/tex}{tex}\\Rightarrow{/tex}a+8d=17....(ii)Subtracting (i) from (ii), we get,5d=10{tex}\\Rightarrow{/tex}d=2{tex}\\Rightarrow{/tex}a=7-3(2)=7-6=1{tex}\\therefore{/tex}{tex}S _ { n } = \\frac { n } { 2 } [ 2 a + ( n - 1 ) d ]{/tex}{tex}= \\frac { n } { 2 } [ 2 ( 1 ) + ( n - 1 ) 2 ]{/tex}{tex}= \\frac { n } { 2 } [ 2 + 2 n - 2 ]{/tex}={tex}\\frac{n}{2}{/tex}{tex}\\times{/tex}2n=n{tex}^2{/tex}