Question

Please give reason for each step......​

Answer 1

Step-by-step explanation:

Given, mth term of AP is n and nth term is m.

To prove, rth term of AP is (m+n-r)

Prove;

The general term of an AP an = a+(n-1)d

n = a+(m-1)d

a = n-(m-1)d .......(i)

m = a+(n-1)d

a = m-(n-1)d ....... (ii)

Equating equation (i) and (ii) since a is same in an AP. n-(m-1)d= m-(n-1)d

n - dm+d = m-dn+d

n - dm = m-dn

dn-dm = m-n

d(n-m) = [-(n-m)]

d = [-(n-m)]/(n-m)

d = -1

Putting d = -1 in equation (i)

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

= n+m-1

Putting a= n+m-1 and d = -1 in general term of AP.

rth term = a+(r-1)d

= n+m-1+(r-1)(-1)

= n+m-1-r+1

= m+n-r

HENCE PROVED

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