Sum of first 20 terms of AP is 400 and sum of first 40 terms is 1600 find sum of first 10 terms

Sum of first 20 terms of AP is 400 and sum of first 40 terms is 1600 f

1.	Sum of first 20 terms of AP is 400 and sum of first 40 terms is 1600 find sum of first 10 terms
Answer» Let the first term be a and common difference be d and sum of 20 terms be S20\xa0{tex}S _ { n } = \\frac { n } { 2 } ( 2 a + (n -1) d ){/tex}{tex}S _ { 20 } = \\frac { 20 } { 2 } ( 2 a + (20 -1) d ){/tex}{tex}S _ { 20 } = \\frac { 20 } { 2 } ( 2 a + 19 d ){/tex}or,\xa0{tex}400 = \\frac { 20 } { 2 } ( 2 a + 19 d ){/tex}or, 400 = 10 [2a + 19d]or 2a + 19d = 40 .......... (i)Also, {tex}S _ { 40 } = \\frac { 40 } { 2 } ( 2 a + (40 -1) d ){/tex}S40 = {tex}\\frac{40}{2}{/tex}(2a + 39d)or, 1600 = 20[2a + 39d]or 2a + 39d = 80 ........... (ii)Subtract (i) and (ii), we get(2a + 39d) - (2a + 19d) = 80 - 402a + 39d - 2a - 19d = 4020d = 40d = 2Put d = 2 in (i)2a + 19d = 402a + 19(2) = 402a + 38 = 402a = 40 - 382a = 2a = 1a = 1 and d = 2{tex}S _ { 10 } = \\frac { 10 } { 2 } [ 2 \\times 1 + ( 10 - 1 ) ( 2 ) ]{/tex}= 5 [2 + 9 {tex}\\times{/tex}\xa02]= 5 [2 + 18]= 5 {tex}\\times{/tex}\xa020= 100