Class 10

Math

All topics

Sequences and Series

The sum of the first $9$ terms of an $A.P$ is $81$ and the sum of it's first $20$ terms is $400.$ Find the first term, the common difference and the sum upto $15th$ term.

- $a=1,d=2,$ $S_{15}=235$
- $a=3,d=4,S_{15}=215$
- $a=5,d=3,S_{15}=205$
- None of these

Given sum of $9$ terms $=81$

$81=29 [2a+(9−1)d]...(1)$

Sum of $20$ terms $=400$

$400=220 [2a+(20−1)d]...(2)$

$⇒81=29 [2a+8d]$

$⇒400=29 [2a+19d]$

$⇒162=18a+72d...(3)$

$⇒400=20a+190d....(4)$

Multiply eq$(3)$ by $20$ and and eq$(4)$ by $18$ and subtract both the equation

$⇒360a+3420d=7200$

$⇒360a+1440d=3240$

$1980d=3690$

$⇒d=2$

Now substitute the $d=2$ in eq$(4)$

$⇒400=20a+190×2$

$⇒400=20a+380$

$⇒20a=20$

$⇒a=1$

Sum of the 15th Term

$S_{15}=215 [2×1+(15−1)2]$

$S_{15}=215 [2+(14)2]$

$S_{15}=215 [2+28]$

$S_{15}=215 [30]$

$S_{15}=15×15=225$

Hence $a=1,d=2,S_{15}=225$