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.