Question

If m times thr mth term of an A.P is equal to ntimes its nth term,find the (m+n)th term of the A.P

Answer 1

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

ma + d(m²-m) = na + d(n²-n)

a(m-n) + d(m²-n² + n-m) = 0

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

Taking (m-n) common and canceling it with zero,

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

Answer 2

Answer:

Step-by-step explanation:Answer:

Let the first term of AP = a

common difference = d

We have to show that (m+n)th term is zero or a + (m+n-1)d = 0

mth term = a + (m-1)d

nth term = a + (n-1) d

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

⇒ am + m²d -md = an + n²d - nd

⇒ am - an + m²d - n²d -md + nd = 0

⇒ a(m-n) + (m²-n²)d - (m-n)d = 0

⇒ a(m-n) + {(m-n)(m+n)}d -(m-n)d = 0

⇒ a(m-n) + {(m-n)(m+n) - (m-n)} d = 0

⇒ a(m-n)  + (m-n)(m+n -1) d  = 0

⇒ (m-n){a + (m+n-1)d} = 0 

⇒ a + (m+n -1)d = 0/(m-n)

⇒ a + (m+n -1)d = 0

Proved!