Sum of first 20 terms of …

Let the first term be a and common difference be d and sum of 20 terms be S₂₀
 {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, {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 - 40
2a + 39d - 2a - 19d = 40
20d = 40
d = 2
Put d = 2 in (i)
2a + 19d = 40
2a + 19(2) = 40
2a + 38 = 40
2a = 40 - 38
2a = 2
a = 1
a = 1 and d = 2
{tex}S _ { 10 } = \frac { 10 } { 2 } [ 2 \times 1 + ( 10 - 1 ) ( 2 ) ]{/tex}
= 5 [2 + 9 {tex}\times{/tex} 2]
= 5 [2 + 18]
= 5 {tex}\times{/tex} 20
= 100