Question

plzz solve the above question 7
plzz help

Answer 1

By the mathematical induction we have to prove this.

⇒ Given that S(n) = 1.2 + 2.22 + 3.23 + ..........+n.2n = (n-1).2n+1 + 2

Let n =1

L.H.S = n.2n = 2

R.H.S = (n-1).2n+1 + 2 = 0 + 2 = 2

L.H.S = R. H .S

∴ S(1) is true.

Let us assume that n = k

S(k) = 1.2 + 2.22 + 3.23 + ..........+k.2k = (k-1).2k+1 + 2 --------→(1)

∴ S(k) is true.

Now we have to prove that n = k+1

Adding (k+1) .2k+1 on equation (1) both sides

⇒ 1.2 + 2.22 + 3.23 + ..........+k.2k + (k+1) .2k+1 = (k-1).2k+1 + 2 + (k+1) .2k+1

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

= 2k.2k+1+ 2

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

∴ s(k+1) is true.

By the mathematical induction it is true for all n∈N.

Answer 2

here is your answer !!!
cheers !!