The fifth term of an arithmetic sequence is equal to 6 and the sum of the first 12 terms is 45.

Find the first term and the common difference.

use of either ${u_n} = {u_1} + (n - 1)d$ or ${S_n} = \frac{n}{2}\left( {2{u_1} + (n - 1)d} \right)$     M1

${u_1} + 4d = 6$    (A1)

$\frac{{12}}{2}(2{u_1} + 11d) = 45$    (A1)

$\Rightarrow 4{u_1} + 22d = 15$

attempt to solve simultaneous equations     M1

$4(6 - 4d) + 22d = 15$

$6d =  - 9 \Rightarrow d =  - 1.5$    A1

${u_1} = 12$    A1

[6 marks]

Most candidates tackled this question through the use of the standard formula for arithmetic series. Others attempted a variety of trial and improvement approaches with varying degrees of success.