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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 81
Total: 106
Let the sum of n, 2n, 3n terms of an A.P. be S1,S2 and S3, respectively, show that S3 = 3(S2–S1).
Solution:  
Let the first term be a and the common difference be d.
According to question
S1 =
n
2
 [2a + (n - 1) d] , S2 =
2n
2
 [2a + (2n - 1) d] , S3 =
3n
2
 [2a + (3n - 1) d]
Now , S2−S1 =
2n
2
 [2a + (2n - 1) d] -
n
2
 [2a + (n - 1) d]
=
n
2
 [(4a + (4n - 2) d) - (2a + (n - 1) d)] =
n
2
 [(4a - 2a) + (4n - 2 - n + 1) d]
=
n
2
 [2a + (3n - 1) d] =
1
3
[
3n
2
[2a+(3n−1)d]]
⇒ S2−S1 =
1
3
S3
Hence S3 = 3(S2−S1)
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