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Prove that 1^(2) +2^(2)+ ….+n^(2) gt ...

Prove that
`1^(2) +2^(2)+ ….+n^(2) gt (n^(3))/(3) n in N`

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If ninNN , then by principle of mathematical induction prove that, 1^(2)+2^(2)+3^(2)+....+n^(2)gt(n^(3))/(3) .

Using the principle of mathematical induction, prove that 1.3 + 2.3^(2) + 3.3^(2) + ... + n.3^(n) = ((2n-1)(3)^(n+1)+3)/(4) for all n in N .

For all n ge 1 prove that 1^(2)+2^(2)+ 3^(2)+4^(2)+….+n^(2)= (n(n+1)(2n+1))/(6)

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1) . Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N .

Prove that n^n > 1, 3, 5,…………(2n - 1) .

By principle of mathematical induction,prove that 1^(3)+2^(3)+3^(3)+ . . .. +n^(3)=[(n(n+1))/(2)]^(2) for all ninNN

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

Prove that .^(n)C_(1) - (1+1/2) .^(n)C_(2) + (1+1/2+1/3) .^(n)C_(3) + "…" + (-1)^(n-1) (1+1/2+1/3 + "…." + 1/n) .^(n)C_(n) = 1/n

By mathematical induction prove that, 1*2+2*2^(2)+3*2^(3)+ . . .+n*2^(n)=(n-1)*2^(n+1)+2,ninNN .

NCERT BANGLISH-PRINCIPLE OF MATHEMATICAL INDUCTION-EXERCISE - 4.1
  1. Prove that 1^(2) +2^(2)+ ….+n^(2) gt (n^(3))/(3) n in N

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  2. Prove that by using the principle of mathematical induction for all n...

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  3. Prove that by using the principle of mathematical induction for all n...

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