Calculate

(i)     the common difference of the sequence;

(ii)     the first term of the sequence.

The first, second and fifth terms of this arithmetic sequence are the first three terms of a geometric sequence.

Calculate the seventh term of the geometric sequence.

(i)     \({u_1} + d = 30,{\text{ }}{u_1} + 4d = 90,{\text{ }}3d = 90 - 30\;\;\;\)(or equivalent)     (M1)

Note: Award (M1) for one correct equation. Accept a list of at least 5 correct terms.

\((d = ){\text{ }}20\)     (A1)

(ii)     \(({u_1} = ){\text{ }}10\)     (A1)(ft)     (C3)

Note: Follow through from (a)(i), irrespective of working shown if \({u_1} = 30 - {\text{ (their }}d)\;\;\;\)OR\(\;\;\;{u_1} = 90 - 4 \times {\text{ (their }}d{\text{)}}\)

\(({u_7} = ){\text{ }}10({3^{(7 - 1)}}\;\;\;\)OR\(\;\;\;({u_7} = ){\text{ 10}} \times {3^6}\)     (M1)(A1)(ft)

Note: Award (M1) for substituted geometric sequence formula, (A1)(ft) for their correct substitutions.

OR

\(10;{\text{ }}30;{\text{ }}90;{\text{ }}270;{\text{ }}810;{\text{ }}2430;{\text{ }}7290\)     (M1)(A1)(ft)

Note: Award (M1) for a list of at least 5 consecutive terms of a geometric sequence, (A1)(ft) for terms corresponding to their answers in part (a).

\( = 7290\)     (A1)(ft)     (C3)

Note: Follow through from part (a).

Part (a) was answered correctly by many candidates, but working using equations was rarely seen. A “trial and error” method, based upon a list of terms was the most seen method.

In part (b) many found the correct answer, but many others did not. Some gave the seventh term of the arithmetic sequence, some gave a term of an incorrect order and some a completely incorrect answer. Finding the correct ratio was the most common problem. Often repeated multiplication was used to find the answer, but also the formula for the nth term of a geometric sequence was used. Several did not use the correct three terms from the question.