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Question

Assertion :If Sn=nC1nC22+nC33.....+(1)n1ncnn & Tn=1+12+13+...+1n, then Sn=Tnn Reason: Sn+1Sn=Tn+1Tn

A
Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
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B
Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
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C
Assertion is true but Reason is false,
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D
Assertion is false but Reason is true.
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Solution

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
Since S1=1C1=1 and T1=1S1=T1
Thus if the reason is true and if Sn=Tn for some n.
then Sn+1=Tn+1Sn=Tn for all n
Thus assertion is true if reason is true.
Let us prove reason
Tn+1Tn=1n+1
Sn+1Sn=n+1C1n+1C22+n+1C33....+(1)n.n+1Cnn
+(1)n.n+1Cn+1n+1(nC1nC23+nC33.....+(1)n1nCnn)
=(n+1C1nC1)12(n+1C2nC1)+13(n+1C3nC3)
....+(1)n1n(n+1CnnCn)+(1)nn+1
=nC0nC12+nC23.....+(1)nnCnn+1
=10(1x)ndx=1n+1
Thus reason is true.

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