Question

if in an ap the sum of first m terms is equal to n and the sum of first n terms is m, prove that the sum of first (m+n) terms is -(m+n)

Answer 1

Let a be the first term and d be c.d. of the A P .Then

Sm=n

n= m/2{2a+ (m-1)d}

2n= 2am+ m( m-1)d. ........(1)

and

Sn= m

m= n/2{2a+(n-1)d}

2m= 2an+ n(n-1)d. ...........(2)

Subtracting eq.(2)- (1), we get

2a(m-1)+{m(m-1)- n(n-1)}d=2n-2m

2a(m-n)+{(m^2-n^2)-(m-n)}d= -2(m-n)

2a +(m+n-1) d = -2.

[On dividing both sides by ( m-n)]………(3)

Now,

Sm+n=m+n/2{2a +(m+n-1)d}

Sm+n=m+n/2(-2) ………[using (3)]

Sm+n=-(m+n)

Hence Proved

Answer 2

Answer:

Step-by-step explanation:

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