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Date	May 2018	Marks available	6	Reference code	18M.1.SL.TZ2.S_7
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Find	Question number	S_7	Adapted from	N/A

Question

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

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]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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22M.2.SL.TZ1.2b.ii:
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22M.2.SL.TZ2.2a.ii:
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22M.2.SL.TZ2.2a.i:
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22M.2.SL.TZ1.2a.i:
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17M.1.AHL.TZ2.H_3b:
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16N.1.SL.TZ0.S_9b:
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19M.1.AHL.TZ2.H_1:
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18M.2.SL.TZ2.T_4c:
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18M.2.SL.TZ2.T_4e:
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17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
17M.1.SL.TZ2.T_5b:
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18M.2.SL.TZ2.T_4b:
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18M.1.SL.TZ1.T_7a.i:
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19M.2.SL.TZ2.T_4d.iii:
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19M.2.SL.TZ1.T_5a:
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17N.1.SL.TZ0.S_2c:
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19N.2.SL.TZ0.T_3b:
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Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the seventh day.
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16N.2.AHL.TZ0.H_12e:
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16N.2.AHL.TZ0.H_12d:
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Write down the common difference, $d$ .
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19N.2.SL.TZ0.T_3e:
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18M.2.SL.TZ1.S_7b:
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19M.2.SL.TZ2.T_4c:
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19M.2.SL.TZ2.S_10d:
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Find the largest value of $a$ such that ${S_\infty } < F$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
20N.2.SL.TZ0.T_5a:
Find $h$ , the height of the tank.
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
21M.1.SL.TZ1.8a.i:
Charlie ran on day $20$ of his fitness programme.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
19N.2.SL.TZ0.T_3a:
Find the number of triangular panes in the $12 th$ level.
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Find the total cost of buying 2 tickets in each of the first 16 rows.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
21M.2.SL.TZ2.3a.ii:
the total number of seats in the concert hall.
21M.2.SL.TZ2.3a.i:
the number of seats in the last row.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
19M.2.SL.TZ2.T_4a.ii:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the eleventh day.
18M.1.AHL.TZ2.H_5b:
A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
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Find the value of $n$ such that $u_{n} = 0$ .
16N.2.AHL.TZ0.H_12c:
Write down an expression for ${A_n}$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$ .
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
EXN.3.AHL.TZ0.2a:
Find an expression for the width of $U_{n}$ in centimetres.
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Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
EXN.1.SL.TZ0.12c:
A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
Daniella ran on day $20$ of her fitness programme.
EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
Calculate the percentage increase in applications from the first year to the second year.
21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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