Question

If pth term of an Ap is equal to q and qth term is p,prove that its nth term is (p+q-n)

Answer 1

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Answer 2

Answer:

To prove: nth term of AP is q + p - n.

Given: pth term of AP is q and qth term of Ap is p.

let, a be the first term of the AP and d be the common difference of the AP

We know Formula of  a term of AP,

$a_n=a+(n-1)d$

So,

$a_p=q$

$a+(p-1)d=q$ .....................(1)

$a_q=p$

$a+(q-1)d=p$ .....................(2)

Subtract eqn (2) from (1)

we get,

( p - 1 )d - ( q - 1 ) d = q - p

d ( p - 1 - q + 1 ) = q - p

d ( p - q ) = q - p

d = -1

Putting value of d in eqn (1)

we get

a + ( p - 1 )(-1) = q

a - p + 1 = q

a = q + p - 1

Now,

nth term given by,

$a_n=(q+p-1)+(n-1)(-1)=q+p-1-n+1=q+p-n$

Hence Proved.