Show that A is an arithmetic sequence, stating its common difference d in terms of r.

A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.

METHOD 1

state that \({u_n} = {u_1}{r^{n - 1}}\) (or equivalent)      A1

attempt to consider \({{a_n}}\) and use of at least one log rule       M1

\({\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right| = {\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right| + \left( {n - 1} \right){\text{lo}}{{\text{g}}_2}\left| r \right|\)      A1

(which is an AP) with \(d = {\text{lo}}{{\text{g}}_2}\left| r \right|\) (and 1^st term \({\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right|\))      A1

so A is an arithmetic sequence      AG

Note: Condone absence of modulus signs.

Note: The final A mark may be awarded independently.

Note: Consideration of the first two or three terms only will score M0.

[4 marks]

METHOD 2

consideration of \(\left( {d = } \right){a_{n + 1}} - {a_n}\)      M1

\(\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {{u_{n + 1}}} \right| - {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|\)

\(\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right|\)     M1

\(\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| r \right|\)     A1

which is constant      R1

Note: Condone absence of modulus signs.

Note: The final A mark may be awarded independently.

Note: Consideration of the first two or three terms only will score M0.

attempting to solve \(\frac{3}{{1 - r}} = 4\)     M1

\(r = \frac{1}{4}\)     A1

\(d =  - \,2\)     A1

[3 marks]