Find ${H_2}$, ${H_3}$ and ${H_4}$.

Conjecture a formula for ${H_n}$ in terms of $n$, for $n \in {\mathbb{Z}^ + }$.

Prove by mathematical induction that your formula is valid for all $n \in {\mathbb{Z}^ + }$.

${H_2} = 2{H_1} + 1$    (M1)

$= 3;{\text{ }}{H_3} = 7;{\text{ }}{H_4} = 15$    A1

[2 marks]

${H_n} = {2^n} - 1$    A1

[1 mark]

let ${\text{P}}(n)$ be the proposition that ${H_n} = {2^n} - 1$ for $n \in {\mathbb{Z}^ + }$

from (a) ${H_1} = 1 = {2^1} - 1$     A1

so ${\text{P}}(1)$ is true

assume ${\text{P}}(k)$ is true for some $k \Rightarrow {H_k} = {2^k} - 1$     M1

then ${H_{k + 1}} = 2{H_k} + 1$     (M1)

$= 2 \times ({2^k} - 1) + 1$    A1

$= {2^{k + 1}} - 1$

${\text{P}}(1)$ is true and ${\text{P}}(k)$ is true $\Rightarrow {\text{P}}(k + 1)$ is true, hence ${\text{P}}(n)$ is true for all $n \in {\mathbb{Z}^ + }$ by the principle of mathematical induction     R1

Note:     Only award the R1 if all earlier marks have been awarded.

[5 marks]

This was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.

This was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.

This was a successful question for many candidates with many wholly correct answers seen. The vast majority of candidates were able to answer parts (a) and (b) and most knew how to start part (c). A few candidates were let down by not realising the need for a degree of formality in the presentation of the inductive proof.