Question

Question 7 Prove the following by using the principle of mathematical induction for all n∈N: 1.3 + 3.5 + 5.7 + ... + (2n-1).(2n+1) = [ n(4n^2 + 6n - 1)] / 3

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1.3 + 3.5 + 5.7 + .......+(2n-1)(2n+1) = n(4n²+6n-1)/3
Let p(n): 1.3 + 3.5 + 5.7 + ......+(2n-1)(2n+1) = n(4n²+6n-1)/3

step1:- for n = 1
P(1): 1.3 = 1.(4.1²+6.1-1)/3
= 1.(4+6-1)/3 = 1.3
It's true.

step2:- for n = k
P(k): 1.3 + 3.5 + 5.7 +......+(2k-1)(2k+1) = k(4k²+6k-1)/3 -------(1)

step3:- for n = k+1
P(k+1) :1.3 + 3.5 + 5.7 +....+(2k-1)(2k+1) + (2k+1)(2k+3) = (k+1){4(k+1)²+6(k+1)-1}/3

from eqn (1)
1.3 + 3.5 + 5.7 + .....+(2k-1)(2k+1)+ = k(4k²+6k-1)/3
add (2k+1)(2k+3) both sides ,

1.3 + 3.5 + 5.7 + .........+(2k-1)(2k+1)+(2k+1)(2k+3)= k(4k² + 6k-1)/3 + (2k+1)(2k+3)
= {k(4k²+6k-1)+3(4k²+8k+5)}/3
= (4k³ + 6k²- k + 12k²+24k+15)/3
= (4k³ + 18k² + 23k + 9)/3
= (4k³ + 4k² +14k²+14k+9k+9)/3
= (k+1)(4k²+14k+9)/3
= (k+1) {4(k+1)² +6(k+1)-1}/3

Hence, p(k+1) is true when p(k) is true . from the mathematical induction, statement is true for all natural numbers.