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]