smc-5115-60-Recursive

Proof, EXT2 P2 EQ-Bank 11

A sequence is given by the recursive formula

for

Using mathematical induction to show the formula for the general term of the sequence is

(3 marks)

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Calculus, EXT2 C1 2022 HSC 14b

Let , where "n" is a non-negative integer.

Show that . (1 mark)

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Show that . (2 marks)

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Show that . (2 marks)

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Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all ,
(2 marks)

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Using parts (ii) and (iv) prove that . (1 mark)

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i.

Mean mark (i) 93%.

ii.

♦♦ Mean mark (ii) 28%.

iii.

iv.

♦ Mean mark (iv) 50%.

♦♦ Mean mark (v) 34%.

Proof, EXT2 P2 2022 HSC 13b

The numbers , for integers , are defined as

These numbers satisfy the relation , for . (Do NOT prove this)

Use mathematical induction to prove that , for all integers . (4 marks)

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Mean mark 58%.

Proof, EXT2 P2 2006 HSC 7c

The sequence is given by

and for

Prove by induction that for , , where (4 marks)

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Hence find the limiting value of as (1 mark)

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$$
$$
$$
$$
$$
$$
$$
$$
$$

ii.

Proof, EXT2 P2 2010 HSC 6b

A sequence is defined by

for , with .

Use mathematical induction to prove that

for all . (3 marks)

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♦ Mean mark 36%.