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Question

If 1,2,3.... are first terms; 1,3,5.... are common differences and S1,S2,S3.... are sums of n terms of given p APs; then S1+S2+S3+.....+Sp is equal to

A
np(np+1)2
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B
n(np+1)2
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C
np(p+1)2
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D
np(np1)2
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Solution

The correct option is A np(np+1)2
We know,
a=1+2+3+...+p=p(p+1)2 ......( 1 )
d=1+3+5+...+(2p1)=p2.[2.1+(p1)2]=p2 ......( 2 )
Sp=n2.[2ap+(n1)dp] ......( 3 )

From eq (1),(2),(3):
S1+S2+...+Sp=n2.[2a+(n1)d]

=n2.[2p(p+1)2+(n1).p2]

=n2.[p2+p+np2p2]

=np(np+1)2

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