Question

If m times the term of an A.P. is equal to n times the n" term, then show
(m + n) term of the A.P. is zero.​

Answer 1

Answer:

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Step-by-step explanation:

The general nth term of an AP is a + (n -1)d.

 

From the given conditions,

 

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

=> am + m2d - md = an + n2d - nd

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

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

 

Rejecting the non-trivial case of m=n, we assume that m and n are different.

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

 

The LHS of this equation denotes the (m+n)th term of the AP, which is Zero.

Answer 2

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!