Use mathematical induction to prove the following. Be sure to (a) clearly state the property you are trying to prove: (b) state and prove the basis case: (c) state the inductive hypothesis: (d) state and prove the inductive step, clearly indicating where you used the inductive hypothesis. sigma^n_i = 1 n*n! = 1 + 1! + 2 * 2! + …. + n*n! = (n + 1)! – 1for all n greaterthanorequalto 1
To prove 1.1!+2.2!+3.3!+………………..+n.n!=(n+1)!-1 using induction
Basic step :
Let n=1
then 1.1!=1.1=1
and (1+1)!-1=2!-1=2-1=1
Hence the given function is true for n=1
Induction Step:
Assume that function is true for F(k)
i.e., 1.1!+2.2!+3.3!+………+k.k!=(k+1)!-1;
Now consider F(k+1) = 1.1!+2.2!+3.3!+………+k.k!+(k+1)(k+1)!
= (k+1)!-1 + (k+1)(k+1)! since by above asumption
= (k+1)! (k+1+1) -1
= (k+1)!(k+2) -1
= (k+2)! -1 As (n+1).n = (n+1)!
Therefore hence proved by F(k+1)
Hence by mathematical induction it is proved that
1.1!+2.2!+3.3!+4.4!+………….+n.n! = (n+1)!-1
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