18M.1.SL.TZ2.S_7

pestleMathematicsAISLPaper 118M· sl-1-2-arithmetic-sequences-and-seriessource ↗

An arithmetic sequence has ${u_1} = {\text{lo}}{{\text{g}}_c}\left( p \right)$ and ${u_2} = {\text{lo}}{{\text{g}}_c}\left( {pq} \right)$ , where $c > 1$ and $p,\,\,q > 0$ .

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

Markscheme / solution

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]

Examiners’ report

[N/A]