Question

If the p th term if an A.P is q and the q th term is p, prove that its n th term is (p+q-n)​

Answer 1

Answer:

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Answer 2

$\huge{\underline{\underline{\red{Answer}}}}$

Let the first term be a

and common difference is d

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Then,

Its pth term is q

》a+(p-1)d=q

》a+pd-d=q

》a=q-pd+d -------i

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Also given,

Its qth term is p

》a+(q-1)d=p

》a+dq-d=p

》 a= p-dq+d------ii

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From i and ii we get,

q+d-pd=p-dq+d

》q-p=pd-dq .....(cancelling d from both side)

》q-p= d(p-q)

》-1(p-q)=d(p-q)

》${\boxed{\pink{d=\:-1}}}$----iii

from i we know,

a=q+d-pd

》a= q-1+p.....( d= -1, from iii)--------iv

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Thus nth term is

a+(n-1)d

=q-1+p+(n-1)d....(putting value of a from iv)

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

=q-1+p-n+1

=q+p-n

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$\huge{\mathfrak{\bf{\blue{Hence\:Proved}}}}$

$\underline{\boxed{\pink{nth\:term\:is\:(q+p-n)}}}$