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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 $u_{1} = lo g_{c} (p)$ ${u_1} = {\text{lo}}{{\text{g}}_c}\left( p \right)$ and $u_{2} = lo g_{c} (p q)$ ${u_2} = {\text{lo}}{{\text{g}}_c}\left( {pq} \right)$ , where $c > 1$ $c > 1$ and $p, q > 0$ $p,\,\,q > 0$ .

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

Markscheme

METHOD 1 (finding $u_{1}$ ${u_1}$ and d)

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

attempt to find $u_{1}$ ${u_1}$ or d using $lo g_{c} c^{k} = k$ ${\text{lo}}{{\text{g}}_c}\,{c^k} = k$ (M1)
eg $lo g_{c} c$ ${\text{lo}}{{\text{g}}_c}\,c$ , $3 lo g_{c} c$ ${\text{3}}\,{\text{lo}}{{\text{g}}_c}\,c$ , correct value of $u_{1}$ ${u_1}$ or d

$u_{1}$ ${u_1}$ = 2, d = 3 (seen anywhere) (A1)(A1)

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

$20 \sum n = 1 u_{n}$ $\sum\limits_{n = 1}^{20} {{u_n}}$ = 610 A1 N2

METHOD 2 (expressing S in terms of c)

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

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

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

correct working (A1)

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

$20 \sum n = 1 u_{n}$ $\sum\limits_{n = 1}^{20} {{u_n}}$ = 610 A1 N2

METHOD 3 (expressing S in terms of c)

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

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

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

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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19M.2.AHL.TZ2.H_7:
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22M.2.SL.TZ1.2b.ii:
$l_{n}$ .
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18M.1.SL.TZ1.T_7a.ii:
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16N.1.AHL.TZ0.H_6a:
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22M.2.SL.TZ1.2a.ii:
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22M.2.SL.TZ1.2b.i:
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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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21N.2.SL.TZ0.2b.i:
Write down the common ratio of the sequence.
17M.1.AHL.TZ2.H_3b:
the value of $r$ ;
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:
Find the tea-shop’s profit during the 11th week.
18M.2.SL.TZ2.T_4e:
In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

Find the value of m.
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:
Calculate the café’s total profit for the first 12 weeks.
16N.2.AHL.TZ0.H_12a:
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18M.1.SL.TZ1.T_7a.i:
For that day find how much weight was added after each lift.
19M.2.SL.TZ2.T_4d.iii:
Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ days.
19M.2.SL.TZ2.T_4b:
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17N.2.SL.TZ0.T_2a.ii:
Write down the distance Rosa runs in the $n$ th training session.
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18M.2.SL.TZ1.S_7a:
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19M.2.SL.TZ1.T_5c:
Calculate, in years, when the bicycle value will be less than 50 USD.
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Find the value of $k$ .
SPM.1.SL.TZ0.2a:
Write down the value of the common difference, $d$
19M.2.SL.TZ1.T_5a:
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17N.1.SL.TZ0.S_2c:
Find the sum of the first ten terms.
16N.1.AHL.TZ0.H_6b:
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17N.2.SL.TZ0.T_2d:
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17M.1.AHL.TZ1.H_7a:
find the value of $d$ .
17M.1.SL.TZ1.T_5b:
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Find the maximum number of rungs in this 1050 cm long ladder.
16N.1.AHL.TZ0.H_6c:
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18N.2.SL.TZ0.S_5:
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19N.2.SL.TZ0.T_3b:
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18N.1.SL.TZ0.S_3a:
Find u₈.
16N.1.SL.TZ0.S_9e:
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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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Show that the tank will never be completely filled using this pump.
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The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ th day. Find the value of $k$ .
19N.2.SL.TZ0.T_3e:
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Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest $m^{2}$ .
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Find the amount of antibiotic, in mg, that Ted receives on the fifth day.
18M.2.SL.TZ2.T_4a:
Find the café’s profit during the 11th week.
19M.2.SL.TZ2.S_10b:
Find the exact value of ${S_k}$ .
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Calculate the total distance Carlos runs in the first year.
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Show that there will be approximately 2645 fish in the lake at the start of 2020.
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Find $d$ , giving your answer as an integer.
19N.2.SL.TZ0.T_3c:
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16N.2.AHL.TZ0.H_12b:
(i) Write down a similar expression for ${A_3}$ and ${A_4}$ .

(ii) Hence show that the amount in Mary’s account the day before she turned 10 years old is given by $251({1.004^{120}} - 1)x$ .
18M.2.SL.TZ1.S_7b:
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19M.2.SL.TZ2.T_4c:
Calculate the total amount of the drug, in mg, that she receives.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
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.
SPM.1.SL.TZ0.2c:
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.
20N.2.SL.TZ0.T_5e.i:
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.
19M.2.SL.TZ1.T_5b:
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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