Question

If in an A.P. the sum of m terms is equal to n and the sum of n terms is equal to m,then prove that the sum of (mn) is 1/2(mn +1)

Answer 1

The proof is as follows:

Step 1:

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

Sm=n

Step 2:

n= m/2{2a+ (m-1) d}  

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

And

Step 3:

S n= m

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

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

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

Step 4:

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

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,

Step 5:

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

Sm + n = m + n / 2 (-2) ………[using (3)]

Sm + n= - ( m + n)

Hence Proved.