To Prove:
1 x 21 + 2 x 22 + 3 x 23 + …. + n x 2n =(n – 1)2n+1 + 2.
Let us prove this question by principle of mathematical induction (PMI)
Let P(n): 1 x 21 + 2 x 22 + 3 x 23 + …. + n x 2n
For n = 1
LHS = 1 × 2 = 2
RHS = (1 - 1) × 2(1 + 1) + 2
= 0 + 2 = 2
Hence, LHS = RHS
P(n) is true for n 1
Assume P(k) is true
1 x 21 + 2 x 22 + 3 x 23 + …. + k x 2k = (k – 1) x 2k+1 + 2.....(1)
We will prove that P(k + 1) is true
1 x 21 + 2 x 22 + 3 x 23 + (k x 1) x 2k + 1 = ((k + 1) - 1) x 2(k+1) + 1 + 2
1 x 21 + 2 x 22 + 3 x 23 + (k x 1) x 2k + 1 = (k) x 2k+2 + 2
1 x 21 + 2 x 22 + 3 x 23 + k2k + (k x 1) x 2k + 1 = (k) x 2k+2 + 2.....(2)
We have to prove P(k + 1) from P(k), i.e. (2) from (1)
From (1)
1 x 21 + 2 x 22 + 3 x 23+ k x 2k = (k – 1) x 2k+1 + 2
Adding (k + 1) x 2k + 1 both sides,
(1 x 21 + 2 x 22 + 3 x 23+ k x 2k) + (k + 1) x 2k + 1 = (k - 1) x 2k + 1 + 2 + (k + 1) x 2k + 1
= k x 2k + 1 - 2k + 1 + + k x 2k + 1 + 2k + 1
= 2k x 2k + 1 + 2
= k x 2k + 2 + 2
(1 x 21 + 2 x 22 + 3 x 23+ k x 2k) + (k + 1) x 2k + 1 = k x 2k + 2 + 2
Which is the same as P(k + 1)
Therefore, P (k + 1) is true whenever P(k) is true
By the principle of mathematical induction, P(n) is true for
Where n is a natural number Put k = n - 1
(1 x 21 + 2 x 22 + 3 x 23) + n x 2n = (n – 1) x 2n+1 + 2
Hence proved.