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Question

State the whether given statement is true or false
For a positive integer n,let S(n)=1+12+13+14+.....+12n−1. Then prove that S(100)<100.

A
True
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B
False
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Solution

The correct option is A True
We have,
S(n)=1+12+13+14+.....+12n1

S(n)=1+(12+13)+(14+15+16+17)+(18+19+.......)+.....+12n1

S(n)=1+(12+1221)+(14+15+16+1231)+(18+19+.......+1241)+.....+12n1

Since,
(12+13)<1
(14+15+16+17)<1
12n1<1

So,
S(n)=1+1+1+......+1 n(times)
S(n)<n

Therefore,
S(100)<100

Hence, this is the answer.

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