I have a question from arithmetic progression chapter in which we are to find the first term and the common difference of an AP, if the sum of first 9 terms of an AP is 81 and that of its first 20 terms is 400.

Book – RS Aggarwal, Class 10, Chapter 5C, question no 28

## Sum of the term Sn=n/2[2×a+(n−1)d]

Given sum of 9 terms =81

81=9/2[2a+(9−1)d]...(1)

Sum of 20 terms =400

400=20/2[2a+(20−1)d]...(2)

⇒81=9/2[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