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p(n) : n(n + 1) is even
p(3) : 3.(3 + 1) is even
p(n) : n3 + n is divisible by 3
p(3) : 33 + 3 is divisible by 3
⟹ p(3) : 30 is divisible by 3.
∴ p(3) is true
Now,
p(4) : 43 + 3 = 67 is divisible by 3
since, 67 is not divisible by 3
so, p(4) is not true
p(n) : 2n ≥ 3n
Given that p(r) is true
⟹ 2r ≥ 3r
Multiplying both the sides by 2,
2.2r ≥ 2.3r
2r+1 ≥ 6r
≥ 3r + 3r
≥ 3 + 3r, [since 3r ≥ 3 ⟹ 3r + 3r ≥ 3 + 3r]
2r+1 ≥ 3(r + 1)
⟹ p(r + 1) is true
Here, p(n) : n2 + n is even
Given, p(r) is true
⟹ r2 + r is even
⟹ r2+ r = 2λ ---- (i)
(r + 1)2 + (r + 1)
= r2 + 2r + 1 + r + 1
= (r2 + r) + 2r + 2
= 2λ + 2r + 2 [Using equation (1)]
= 2(λ + r + 1)
= 2µ
⟹ (r + 1)2 + (r + 1) is even
p(n) : n2 - n + 41 is prime
p(1) : 1 - 1 + 41 is prime
⟹ p(1) : 41 is prime
∴ p(1) is true.
p(2) : 22 - 2 + 41 is prime
⟹ p(2) : 43 is prime
∴ p(2) is true.
p(3) : 32 - 3 + 41 is prime
⟹ p(3): 47 is prime
∴ p(3) is true.
p(41): (41)2 - 41 + 41 is prime
p(41): (41)2 is prime
⟹ p(41) is not true
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Chapter 12: Mathematical Induction –...