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Question

# Assertion :If Sn=nC1−nC22+nC33−.....+(−1)n−1ncnn & Tn=1+12+13+...+1n, then Sn=Tn∀n Reason: Sn+1−Sn=Tn+1−Tn

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=1⇒S1=T1Thus if the reason is true and if Sn=Tn for some n.then Sn+1=Tn+1⇒Sn=Tn for all nThus assertion is true if reason is true.Let us prove reasonTn+1−Tn=1n+1Sn+1−Sn=n+1C1−n+1C22+n+1C33−....+(−1)n.n+1Cnn+(−1)n.n+1Cn+1n+1−(nC1−nC23+nC33−.....+(−1)n−1nCnn)=(n+1C1−nC1)−12(n+1C2−nC1)+13(n+1C3−nC3)−....+(−1)n−1n(n+1Cn−nCn)+(−1)nn+1=nC0−nC12+nC23−.....+(−1)nnCnn+1=∫10(1−x)ndx=1n+1Thus reason is true.

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