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