# Answered Essay: Use mathematical induction to prove the following. Be sure to (a) clearly state the property you are trying to

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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