Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).

Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).

valid approach involving addition or subtraction       M1
eg  \({u_2} = {\text{lo}}{{\text{g}}_c}\,p + d,\,\,{u_1} - {u_2}\)

correct application of log law      A1
eg  \({\text{lo}}{{\text{g}}_c}\left( {pq} \right) = {\text{lo}}{{\text{g}}_c}\,p + {\text{lo}}{{\text{g}}_c}\,q,\,\,{\text{lo}}{{\text{g}}_c}\left( {\frac{{pq}}{p}} \right)\)

\(d = {\text{lo}}{{\text{g}}_c}\,q\)    AG N0

[2 marks]

METHOD 1 (finding \({u_1}\) and d)

recognizing \(\sum { = {S_{20}}} \) (seen anywhere)      (A1)

attempt to find \({u_1}\) or d using \({\text{lo}}{{\text{g}}_c}\,{c^k} = k\)     (M1)
eg  \({\text{lo}}{{\text{g}}_c}\,c\), \({\text{3}}\,{\text{lo}}{{\text{g}}_c}\,c\), correct value of \({u_1}\) or d

\({u_1}\) = 2, d = 3 (seen anywhere)      (A1)(A1)

correct working     (A1)
eg  \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)

\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610     A1 N2

METHOD 2 (expressing S in terms of c)

recognizing \(\sum { = {S_{20}}} \) (seen anywhere)      (A1)

correct expression for S in terms of c      (A1)
eg  \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)

\({\text{lo}}{{\text{g}}_c}\,{c^2} = 2,\,\,\,{\text{lo}}{{\text{g}}_c}\,{c^3} = 3\)  (seen anywhere)     (A1)(A1)

correct working      (A1)

eg  \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)

\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610     A1 N2

METHOD 3 (expressing S in terms of c)

recognizing \(\sum { = {S_{20}}} \) (seen anywhere)      (A1)

correct expression for S in terms of c      (A1)
eg  \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)

correct application of log law     (A1)
eg  \(2\,{\text{lo}}{{\text{g}}_c}\,{c^2} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^4},\,\,19\,{\text{lo}}{{\text{g}}_c}\,{c^3} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}},\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^2}} \right)}^2} + \,\,{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^3}} \right)}^{19}}} \right),\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{c^4} + \,{\text{lo}}{{\text{g}}_c}\,{c^{57}}} \right),\,\,10\left( {{\text{lo}}{{\text{g}}_c}\,{c^{61}}} \right)\)

correct application of definition of log      (A1)
eg  \({\text{lo}}{{\text{g}}_c}\,{c^{61}} = 61,\,\,{\text{lo}}{{\text{g}}_c}\,{c^4} = 4,\,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}} = 57\)

correct working     (A1)
eg  \({S_{20}} = \frac{{20}}{2}\left( {4 + 57} \right),\,\,10\left( {61} \right)\)

\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610     A1 N2

[6 marks]