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Question

Check whether the statement is true or false then true give reason
1+1(1+2)+1(1+2+3)+...+1(1+2+3+...n)=2n(n+1)

A
True
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B
False
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Solution

The correct option is A True

Let P(n) be the given statement, then by principle of mathematical induction,

P(n)=1+11+2+11+2+3++11+2+3++n=2nn+1

Let n=1,

P(1)=2×11+1

=1

So, the statement is true.

Let n=k,

P(k)=1+11+2+11+2+3++11+2+3++k=2kk+1

It is to prove that P(k+1) is true for n=k+1.

Consider,

1+11+2+11+2+3++11+2+3++k+11+2+3++k+(k+1)

=(1+11+2+11+2+3++11+2+3++k)+11+2+3++k+(k+1)

=2kk+1+11+2+3++k+(k+1)

=2kk+1+1(k+1)(k+1+1)2 (n=n(n+1)2)

=2kk+1+2(k+1)(k+1+1)

=2kk+1(k+1k+2)

=2kk+1(k2+2k+1k+2)

=2(k+1)2(k+1)(k+2)

=2(k+1)k+2

=2nn+1

This shows that P(k+1) is true when P(k) is true.

Hence, P(n) is true for all natural numbers.


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