Write down the value of

(i)     ${S_1}$;

(ii)     ${S_2}$.

The ${n^{{\text{th}}}}$ term of the arithmetic sequence is given by ${u_n}$.

Show that ${u_2} = 9$.

The ${n^{{\text{th}}}}$ term of the arithmetic sequence is given by ${u_n}$.

Find the common difference of the sequence.

The ${n^{{\text{th}}}}$ term of the arithmetic sequence is given by ${u_n}$.

Find ${u_{10}}$.

The ${n^{{\text{th}}}}$ term of the arithmetic sequence is given by ${u_n}$.

Find the lowest value of $n$ for which ${u_n}$ is greater than $1000$.

The ${n^{{\text{th}}}}$ term of the arithmetic sequence is given by ${u_n}$.

There is a value of $n$ for which

$${u_1} + {u_2} +  \ldots  + {u_n} = 1512.$$

Find the value of $n$.

(i)     ${S_1} = 7$     (A1)

(ii)     ${S_2} = 16$     (A1)

$({u_2} = ){\text{ }}16 - 7 = 9$     (M1)(AG)

Note: Award (M1) for subtracting 7 from 16. The 9 must be seen.

OR

$16 - 7 - 7 = 2$

$({u_2} = ){\text{ }}7 + (2 - 1)(2) = 9$     (M1)(AG)

Note: Award (M1) for subtracting twice $7$ from $16$ and for correct substitution in correct arithmetic sequence formula.

The $9$ must be seen.

Do not accept: $9 - 7 = 2,{\text{ }}{u_2} = 7 + (2 - 1)(2) = 9$.

${u_1} = 7$     (A1)(ft)

$d = 2{\text{ }}( = 9 - 7)$     (A1)(ft)(G2)

Notes: Follow through from their ${S_1}$ in part (a)(i).

$7 + 2 \times (10 - 1)$     (M1)

Note: Award (M1) for correct substitution in the correct arithmetic sequence formula. Follow through from their parts (a)(i) and (c).

$= 25$     (A1)(ft)(G2)

Note: Award (A1)(ft) for their correct tenth term.

$7 + 2 \times (n - 1) > 1000$     (A1)(ft)(M1)

Note: Award (A1)(ft) for their correct expression for the ${n^{{\text{th}}}}$ term, (M1) for comparing their expression to $1000$. Accept an equation. Follow through from their parts (a)(i) and (c).

$n = 498$     (A1)(ft)(G2)

Notes: Answer must be a natural number.

$6n + {n^2} = 1512\;\;\;$OR$\;\;\;\frac{n}{2}\left( {14 + 2(n - 1)} \right) = 1512\;\;\;$OR

${S_n} = 1512\;\;\;$OR$\;\;\;7 + 9 +  \ldots  + {u_n} = 1512$     (M1)

Notes: Award (M1) for equating the sum of the first $n$ terms to $1512$. Accept a sum of at least the first 7 correct terms.

$n = 36$     (A1)(G2)

Note: If $n = 36$ is seen without working, award (G2). Award a maximum of (M1)(A0) if $- 42$ is also given as a solution.