The geometric sequence u1, u2, u3, … has common ratio r.Consider the sequence A = { a n = lo g 2 | u n | : n ∈ Z + } .

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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 finalAmark may be awarded independently.

Note:Consideration of the first two or three terms only will scoreM0.

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

$r=\frac{1}{4}$ A1

$d=-2$ A1

[3 marks]