HL mathematics mathematical induction past paper exam questions

Induction Test Review 1.

(a)

IB Math HL

Consider the following sequence of equations.

1 (1 × 2 × 3), 3 1 1 × 2 + 2 × 3 = (2 × 3 × 4), 3 1 1 × 2 + 2 × 3 + 3 × 4 = (3 × 4 × 5), 3 .... . 1×2=

(i)

Formulate a conjecture for the nth equation in the sequence.

(ii)

Verify your conjecture for n = 4.

(2)

(b)

A sequence of numbers has the nth term given by un = 2n + 3, n  +. Bill conjectures that all members of the sequence are prime numbers. Show that Bill’s conjecture is false. (2)

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1

Induction Test Review (c)

Use mathematical induction to prove that 5 × 7n + 1 is divisible by 6 for all n 

IB Math HL +

.

(6) (Total 10 marks)

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2

Induction Test Review 2.

(a)

IB Math HL

Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... . (3)

(b)

Use mathematical induction to prove that for n a + ar + ar2 + ... + arn–1 =



+

,



a 1 r n . 1 r (7) (Total 10 marks)

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3

Induction Test Review 3.

IB Math HL

(a)

The sum of the first 4 terms of an arithmetic series is 42. The sum of the first 9 terms of an arithmetic series is 162. Find the first term and the common difference.

(b)

The sum of the first 2 terms of a geometric series is geometric series is

15 . The sum of the first 4 terms of a 4

255 . If all of its terms are positive, find the first term and the common 64

ratio.

(c)

Let the arithmetic sequence in part (a) be an and the geometric sequence in part (b) be g n . Let a new series, cn 

an n 1 . Show that cn   n  1 4  . gn

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Induction Test Review n

(d)

Prove that

a r 1

r

IB Math HL



3n  n  3 . 2

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5

Induction Test Review

4.

A sequence is defined by u1  1 and un  un1   2n  1 for all n 

IB Math HL 

. By finding un

for n  2,3, 4 , conjecture a formula for un in terms of n only. Prove that your conjecture is true using the principle of mathematical induction.

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