By using the principle of mathematical induction, prove that2.7 +3.55

1.	By using the principle of mathematical induction, prove that2.7 +3.55 is divisible by 24, Vn N
Answer» Letp(n) be the statement given by p(1)= 2.7+3.5-524 which is divisible by 12 ∴p(1) is true Letp(m) : 2.7m+ 3.5m– 5 is divisible by 24 be true ⇒ 2.7m+ 3.5m– 5 = 24λ ;λ∈N ⇒ 3.5m= 24λ + 5 – 2.7m... (1) p(m+ 1) : 2.7m+ 1+ 3.5m+ 1– 5 = 2.7m+ 1+ (3.5m) 5 – 5 = 2.7m+ 1+ (24λ + 5 – 2.7m) 5 – 5 (from (1)) = 2.7m+ 1+ 120λ + 25 – 10.7m– 5 = (2.7m+ 1– 10.7m) + 120λ+ 20 = (2 × 7λ7m– 10 × 7m) + 120λ+ 24 – 4 = (14 – 10)7m– 4 + 24 (5λ+ 1) = 4 (7m– 1) + 24 (5λ+ 1) = 4 × 6µ + 24(5λ+ 1) (∵ 7m– 1 is a multiple of 6 for allm∈N∴ 7m– 1 = 6µ, µ∈N) = 24 (µ +5λ+ 1) which is divisible by 24 ∴p(m+ 1) is true. Hence by principle of mathematical inductionp(n) is true for alln∈N

By using the principle of mathematical induction, prove that2.7 +3.55 is divisible by 24, Vn N

Answer»

Letp(n) be the statement given by

p(1)= 2.7+3.5-524 which is divisible by 12

∴p(1) is true

Letp(m) : 2.7m+ 3.5m– 5 is divisible by 24 be true

⇒ 2.7m+ 3.5m– 5 = 24λ ;λ∈N

⇒ 3.5m= 24λ + 5 – 2.7m... (1)

p(m+ 1) : 2.7m+ 1+ 3.5m+ 1– 5

= 2.7m+ 1+ (3.5m) 5 – 5

= 2.7m+ 1+ (24λ + 5 – 2.7m) 5 – 5 (from (1))

= 2.7m+ 1+ 120λ + 25 – 10.7m– 5

= (2.7m+ 1– 10.7m) + 120λ+ 20

= (2 × 7λ7m– 10 × 7m) + 120λ+ 24 – 4

= (14 – 10)7m– 4 + 24 (5λ+ 1)

= 4 (7m– 1) + 24 (5λ+ 1)

= 4 × 6µ + 24(5λ+ 1) (∵ 7m– 1 is a multiple of 6 for allm∈N∴ 7m– 1 = 6µ, µ∈N)

= 24 (µ +5λ+ 1) which is divisible by 24

∴p(m+ 1) is true.

Hence by principle of mathematical inductionp(n) is true for alln∈N