Chapter 12: Mathematical Induction – Exercise 12.1

Mathematical Induction – Exercise 12.1 – Q.1

p(n) : n(n + 1) is even

p(3) : 3.(3 + 1) is even

 

Mathematical Induction – Exercise 12.1 – Q.2

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

 

Mathematical Induction – Exercise 12.1 – Q.3

p(n) : 2ⁿ ≥ 3n

Given that p(r) is true

⟹ 2^r ≥ 3r

Multiplying both the sides by 2,

2.2^r  ≥ 2.3r

2^r+1 ≥ 6r

≥ 3r + 3r

≥ 3 + 3r,         [since 3r ≥ 3 ⟹ 3r + 3r ≥ 3 + 3r]

2^r+1 ≥ 3(r + 1)

⟹ p(r + 1) is true

 

Mathematical Induction – Exercise 12.1 – Q.4

Here, p(n) : n² + n is even

Given, p(r) is true

⟹ r² + r is even

⟹ r²+ r = 2λ  ---- (i)

Now,

(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(r + 1) is true

 

Mathematical Induction – Exercise 12.1 – Q.5