Question

Question 8 Prove the following by using the principle of mathematical induction for all n∈N: 1.2 + 2.2^2 + 3.2^2 + … + n.2^n = (n – 1) 2^(n+1) + 2

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1.2 + 2.2² + 3.2³ + .......n.2ⁿ = (n-1)2^(n+1) +2
Let P(n): 1.2 + 2.2² + 3.2³ +.......n.2ⁿ = (n-1).2^(n+1) + 2

step1 :- for n = 1
P(1): 1.2 = (1-1).2^(1+1) + 2 = 0 + 2 = 2
which is true .

step2:- for n = k
P(k): 1.2 + 2.2² + 3.2³ + .......+k.2^k = (k-1)2^(k+1) +2 -------(1)

step3:- for n = (k+1)
P(k+1):1.2 +2.2² + 3.2³ + .......+ k.2^k + (k+1)2^(k+1) = (k+1-1)2^(k+2) +2

from equation (1) ,
1.2 + 2.2² + 3.2³ + .......k.2^k = (k-1)2^(k+1) +2
add (k+1)2^(k+1) both sides,
1.2 + 2.2² + 3.2³ + ........k.2^k + (k+1)2^(k+1)= (k-1).2^(k+1) + 2 + (k+1)2^(k+1)
= 2^(k+1){k-1 + k+1 } + 2
= 2^(k+1) .2k
= 2.2^(k+1).k
= [(k+1) - 1] .2^{(k+1)+1]
hence, p(k+1) is true when p(k) is true . from the principle of mathematical induction, statement is true for all real numbers.