SSC (Marathi Semi-English) 10thMaharashtra State Board
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# If the sum of first p terms of an A.P. is equal to the sum of first q terms, then show that the sum of its first (p + q) terms is zero where p ≠ q. - SSC (Marathi Semi-English) 10th - Algebra

#### Question

If the sum of first p terms of an A.P. is equal to the sum of first q terms, then show that the sum of its first (p + q) terms is zero where p ≠ q.

#### Solution

To show : S_(p+q)=0

that is, to show : (p+q)/2(2a+(p+q-1)d)=0

Given that S_p=S_n

Let a be the first term of the AP and d be the common difference

p/2(2a+(p-1)d)=q/2(2a+(q-1)d)

p(2a+(p-1)d)=q(2a+(q-1)d)

2ap+(p-1)dp=2aq+(q-1)dq

2ap-2aq+(p-1)dp-(q-1)dq=0

2a(p-q)+d[p^2-p-q^2+q]=0

2a(p-q)+d[p^2-q^2-1(p-q)]=0

2a(p-q)+d[(p+q)(p-q)-1(p-q)]=0

2a(p-q)+d(p-q)[p+q-1]=0

Dividing throughout by (p - q),since p ≠ q.

2a+(p+q-1)d=0 ... (i)

Using eq (i)

S_(p+q)=(p+q)/2(2a+(p+q-1)d)

S_(p+q)=(p+q)/2xx0

S_(p+q)=0

Hence proved.

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Solution If the sum of first p terms of an A.P. is equal to the sum of first q terms, then show that the sum of its first (p + q) terms is zero where p ≠ q. Concept: Sum of First n Terms of an AP.
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