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Question

If sn=1+12+13+14+....1n,(nN), then s1+s2+s3+s4+......sn=(n+λ)sn+1(n+1). Find the value of λ.

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Solution

s1=1
s2=1+12
s3=1+12+13
s4=1+12+13+14
.........................
sn=1+12+13+14+15+16+......1n1+1n
on adding
s1+s2+.....sn
n+n12+n23+n34+.....n(n2)n1+n(n1)n=n(1+12+13+.....1n)(12+23+34+....n2n1+n1n)
=nsn1+1+1+....1(n1)times+(12+13+14+....1n1+1n)
=nsn(n1)+sn1
=(n+1)snn

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