If m times the mth term of an AP is equal to the n times its nth term show that the (m+n)th term of the ap is zero..

I found its sol in rd Sharma but i am nt able to understand the stpes..

pls explaing evry step...

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it is given that mth term is equal to nth term.therefore,

m am = n an

using an formula a +(n-1)d,

m [a(m-1)d] = n [a + (n-1)d]

Multiply the equation with m and n,

ma + (m2-m)d = na + (n2 - n)d

bring both equations to 1 side,

ma - na + (m2-m)d - (n2-n)d =0

take common factor "a and d" out,

(m-n)a + (m2 - m -n2 +n )d = 0

bring square terms together,

(m-n)a + (m- n2- m - n)d =0

Now, it is in the for a2-b2 write it as (a+b) (a-b)

(m-n)a [(m+n)(m-n) - (m-n)] d =0

take (m-n) out as common,

(m-n) [a+(m+n-1)d] =0

(m-n) can be taken to the other side and it becomes 0,

a+ (m+n-1)d =0

the final equation is obtained from the an formula a+(n-1)d

am + an =0

 

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