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 termÂ Snâ€‹=n/2â€‹[2Ã—a+(nâˆ’1)d]

Given sum ofÂ 9Â termsÂ =81

81=9/2â€‹[2a+(9âˆ’1)d]...(1)

Sum ofÂ 20Â termsÂ =400

400=20â€‹/2[2a+(20âˆ’1)d]...(2)

â‡’81=9â€‹/2[2a+8d]

â‡’400=29â€‹[2a+19d]

â‡’162=18a+72d...(3)

â‡’400=20a+190d....(4)

Multiply eq(3)Â byÂ 20Â and and eq(4)Â byÂ 18Â and subtract both the equation

â‡’360a+3420d=7200

â‡’360a+1440d=3240

1980d=3690

â‡’d=2

Now substitute theÂ d=2Â in eq(4)

â‡’400=20a+190Ã—2

â‡’400=20a+380

â‡’20a=20

â‡’a=1