If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m+n) terms is zero.
Given : Sm = Sn
To Prove : S(m+n) = 0
Formula : Sum of first n terms of an A.P. is
Answer –
Sum of first m terms of an A.P. is
And sum of first n terms of an A.P. is
As per given condition, Sm = Sn
∴ m [2a+(m-1)d] = n [2a+(n-1)d]
∴ 2am + m2d – md = 2an + n2d – nd
∴ 2am + m2d – md – 2an – n2d + nd = 0
∴ 2a (m – n) + (m2 – m - n2 + n)d = 0
∴ 2a (m – n) + [(m2 - n2) – (m – n)]d = 0
∴ 2a (m – n) + [(m – n)(m + n) – (m – n)]d = 0
∴ 2a (m – n) + (m – n) [(m + n) –1] d = 0
∴ 2a (m – n) + (m – n) [m + n –1] d = 0
∴ (m – n) [2a + (m + n –1)d] = 0
∴ 2a + (m + n –1)d = 0 ……….(1)
Now, sum of first (m+n) terms of A.P. is
………from (1)
∴ S(m+n) = 0
Hence, sum of first (m+n) numbers of an A.P. is 0.
Couldn't generate an explanation.
Generated by AI. May contain inaccuracies — always verify with your textbook.
