Question

If amphotin mon of AP is 1 by an and answer term is 1 by and then find the sum of its first mn terms

Answer 1

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10th

Maths

Arithmetic Progressions

Sum of an AP

If mth term of an AP is 1/n...

MATHS

If mth term of an AP is 1/n and nth term is 1/m then find the sum of its first mn terms.

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ANSWER

Let a be first term and d be the common difference of an AP.

mth term=1/n

so,

a

m

=a+(m−1)d=1/n .........(1)

nth term=1/m

a

n

=a+(n−1)d=1/m ........(2)

Subtract (2) from (1)

(m−1)d−(n−1)d=

n

1

−

m

1

d(n−n)=

mn

m−n

d=1/mn .........(3)

From (3) and (1), we get

a+(m−1)×

mn

1

=

n

1

⇒a=

n

1

−

mn

m−1

a=

mn

1

Sum of first k terms =(k/2)[2a+(k−1)d]

S

mn

=mn/2[2(1/mn)+(mn−1)(1/mn)]

=mn/2[1/mn+1]

=(1+mn)/2.