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Home > Mathematics > Extension 1 > Mathematical Induction > HSC and other questions (MANSW)

 

There are various types of questions which can be treated. Here are some examples:

I

1. To prove that 1 + 3 + 5 + . . . + (2n - 1) = n²

2. To prove that

3. Show that 3⁴ⁿ - 1 is divisible by 80 for all positive integral values of n.

4. Show that

5. Show that xⁿ - 1 is divisible by (x - 1).

6. Show that sin (x + 180n) = (-1)ⁿ sin x  for integers n > 0.

7. Show that for

8. Show that n ! > 2ⁿ for n  > 4.

II

Year 12 HSC questions on mathematical induction

1989

(a) By considering the sum of the terms of an arithmetic series, show that
(1 + 2 + 3 + . . . + n)² =

(b) By using the Principle of Mathematical Induction, prove that
1³ + 2³ + 3³ + . . . + n³ = (1 + 2 + 3 + . . . + n)² for all n 1.

1988

Prove by mathematical induction that for n 1,
1³ + 3³ + 5³ + . . . + (2n - 1)³ =

1986

Prove by mathematical induction that
1 x 2⁰ + 2 x 2¹ + 3 x 2² + . . . + n x 2^{n - 1}
= 1 + (n - 1) 2² for all integers n 1.

1985

Use the Principle of Mathematical Induction to prove that 5ⁿ + 2 x (11ⁿ) is a multiple of 3 for all positive integers n.

III

1. The nth term of a series is given by

(a) Find u₅ and u_k+1

(b) Assuming that the sum S_k of the first k  terms of this series is given by the formula

prove that

(c) Explain why the sum of the first n terms of the series is

2. If Sn = 1 x 2 + 2 x 3 + . . . + n (n + 1), use the Principle of Mathematical Induction to show that
for all positive integers n.

3. Write down the formula for the sum of the first n positive odd integers. Explain the method of mathematical induction and use it to prove this formula.

4. Use the method of mathematical induction to show that the sum of the squares of the first n positive integers is

5. Use mathematical induction to prove that, for any positive integer n, 5^{n - 1} - 1 is divisible by 4.

6. (a) Use mathematical induction to prove the identity:

1.2.3.4 + 2.3.4.5 + 3.4.5.6 + . . . + n (n + 1) (n + 2) (n + 3)

= n (n + 1) (n + 2) (n + 3) (n + 4).

(b) Hence state the limit of

as n increases indefinitely.

7. Prove by mathematical induction that:

for all positive integers n.

8. Prove by induction that

for all positive integers n and x 0, 1.

9. (a) Factorise the polynomial 2n² + 7n + 6.

(b) By use of the Principle of Mathematical Induction, prove the relation
6(1²+ 2²+ 3² + . . . + n²) = n (n + 1) (2n + 1 ).

(c) Hence, find the value of the limit

10. Prove by mathematical induction that

11. Use mathematical induction to show that

sinq + sin 2q +sin 3 q+ ...+ sin nq

=

12. The Fibonacci numbers are defined by

Fn = Fn-1 + Fn-2 , n 3,

and F1 = F2 = 1,

where Fn is the nth Fibonacci number.

Prove, by mathematical induction that

F1 + F2 + F3 + S + Fn = Fn-2 - 1.

13. Use mathematical induction to prove that

if

then