Question

If the sum of first P terms of an A P is q and the sum of the first q terms is p . Then show that the sum of the first (P+Q)terms is(- (P+Q))

Answer 1

Answer:

instead of p and q i have used m and n,

Step-by-step explanation:

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)