A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by \({u_{n + 1}} = (1 + {\text{i}}){u_n}\) and \({u_1} = 3\).

(a)     Find the fourth term of the sequence, giving your answer in the form \(x + y{\text{i, }}x,{\text{ }}y \in \mathbb{R}\).

(b)     Find the sum of the first 20 terms of \(\left\{ {{u_n}} \right\}\), giving your answer in the form \(a \times (1 + {2^m})\) where \(a \in \mathbb{C}\) and \(m \in \mathbb{Z}\) are to be determined.

A second sequence \(\left\{ {{v_n}} \right\}\) is defined by \({v_n} = {u_n}{u_{n + k}},{\text{ }}k \in \mathbb{N}\).

(c)     (i)     Show that \(\left\{ {{v_n}} \right\}\) is a geometric sequence.

          (ii)     State the first term.

          (iii)     Show that the common ratio is independent of k.

A third sequence \(\left\{ {{w_n}} \right\}\) is defined by \({w_n} = \left| {{u_n} - {u_{n + 1}}} \right|\).

(d)     (i)     Show that \(\left\{ {{w_n}} \right\}\) is a geometric sequence.

          (ii)     State the geometrical significance of this result with reference to points on the complex plane.

(a)     \(r = 1 + {\text{i}}\)     (A1)

\({u_4} = 3{(1 + {\text{i}})^3}\)     M1

\( =  - 6 + 6{\text{i}}\)     A1

[3 marks]

 

(b)     \({S_{20}} = \frac{{\left( {{{(1 + {\text{i}})}^{20}} - 1} \right)}}{{\text{i}}}\)     (M1)

\( = \frac{{3\left( {{{(2i{\text{)}}}^{10}} - 1} \right)}}{{\text{i}}}\)     (M1)

 

Note:     Only one of the two M1s can be implied. Other algebraic methods may be seen.

 

\( = \frac{{3\left( { - {2^{10}} - 1} \right)}}{{\text{i}}}\)     (A1)

\( = 3{\text{i}}\left( {{2^{10}} + 1} \right)\)     A1

[4 marks]

 

(c)     (i)     METHOD 1

\({v_n} = \left( {3{{(1 + {\text{i}})}^{n - 1}}} \right)\left( {3{{(1 + {\text{i}})}^{n - 1 + k}}} \right)\)     M1

\(9{(1 + {\text{i}})^k}{(1 + i)^{2n - 2}}\)     A1

\( = 9{(1 + {\text{i}})^k}{\left( {{{(1 + i)}^2}} \right)^{n - 1}}\left( { = 9{{(1 + {\text{i}})}^k}{{(2{\text{i}})}^{n - 1}}} \right)\)

this is the general term of a geometrical sequence     R1AG

 

Notes:     Do not accept the statement that the product of terms in a geometric sequence is also geometric unless justified further.

     If the final expression for \({v_n}\) is \(9{(1 + {\text{i}})^k}{(1 + i)^{2n - 2}}\) award M1A1R0.

 

METHOD 2

\(\frac{{{v_{n + 1}}}}{{{v_n}}} = \frac{{{u_{n + 1}}{u_{n + k + 1}}}}{{{u_n}{u_{n + k}}}}\)     M1

\( = (1 + {\text{i}})(1 + {\text{i}})\)     A1

this is a constant, hence sequence is geometric     R1AG

 

Note:     Do not allow methods that do not consider the general term.

 

(ii)     \(9{(1 + {\text{i}})^k}\)     A1

(iii)     common ratio is \({(1 + i)^2}{\text{ }}( = 2i)\) (which is independent of k)     A1

[5 marks]

 

(d)     (i)     METHOD 1

\({w_n}\left| {3{{(1 + i)}^{n - 1}} - 3{{(1 + {\text{i}})}^n}} \right|\)     M1

\( = 3{\left| {1 + i} \right|^{n - 1}}\left| {1 - (1 + {\text{i)}}} \right|\)     M1

\( = 3{\left| {1 + i} \right|^{n - 1}}\)     A1

\(\left( { = 3{{\left( {\sqrt 2 } \right)}^{n - 1}}} \right)\)

this is the general term for a geometric sequence   R1AG

METHOD 2

\({w_n} = \left| {{u_n} - (1 + {\text{i}}){u_n}} \right|\)     M1

\( = \left| {{u_n}} \right|\left| { - {\text{i}}} \right|\)

\( = \left| {{u_n}} \right|\)     A1

\( = \left| {3{{(1 + {\text{i}})}^{n - 1}}} \right|\)

\( = 3{\left| {(1 + {\text{i}})} \right|^{n - 1}}\)     A1

\(\left( { = 3{{\left( {\sqrt 2 } \right)}^{n - 1}}} \right)\)

this is the general term for a geometric sequence     R1AG

 

Note:     Do not allow methods that do not consider the general term.

 

(ii)     distance between successive points representing \({u_n}\) in the complex plane forms a geometric sequence     R1

 

Note:     Various possibilities but must mention distance between successive points.

 

[5 marks]

 

Total [17 marks]

[N/A]