# FPM Arithmetic and Geometric Series (Chapter 5)

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1. (2011/june/Paper02/q1)
Evaluate $\displaystyle\sum_{n=6}^{20}(2 n-3)$ (3 marks)

2. (2012/june/рарег01/q5)
The first four terms of an arithmetic series, $S$, are

$$\log _{a} 2+\log _{a} 4+\log _{a} 8+\log _{a} 16$$

(a) Write down an expression for the $r$ th term of $S$. (1 mark)

(b) Find an expression for the common difference of $S$. (2 marks)

The sum of the first $n$ terms of $S$ is $S_{n}$

(c) Show that $S_{n}=\displaystyle\frac{1}{2} n(n+1) \log _{a} 2$ (2 marks)

The first four terms of a second arithmetic series, $T$, are

$$\log _{a} 6+\log _{a} 12+\log _{a} 24+\log _{0} 48$$

The sum of the first $n$ terms of $T$ is $T_{n}$

(d) Find $T_{n}-S_{n}$ and simplify your answer. (4 marks)

3. $\left(2013 / \mathrm{jan} / \mathrm{Paper01} / \mathrm{q4}\right)$

(a) Show that $\displaystyle\sum_{r=1}^{n}(3 r-4)=\displaystyle\frac{n}{2}(3 n-5)$ (3 marks)

(b) Hence, or otherwise, evaluate $\displaystyle\sum_{r=1}^{50}(3 r-4)$ (2 marks)

Given that $\displaystyle\sum_{r=1}^{h}(3 r-4)=186$

(c) find the value of $n$ (3 marks)

4. $(2013 / \mathrm{jan} / \mathrm{Paper01} / \mathrm{q} 9)$
The sum $S_{n}$ of the first $n$ terms of an arithmetic series is given by $S_{n}=n(2 n+3) .$ The first term of the series is $a$.

(a) Show that $a=5$ (2 marks)

(b) Find the common difference of the series. (3 marks)

(c) Find the 12 th term of the series. (2 marks)

Given that $1+S_{p+4}=2 S_{p}$

(d) find the value of $p$ (4 marks)

5. (2013/june/paper01/q7)
An arithmetic series has first term $a$ and common difference $d$. The $n$th term of the series is $I_{n}$ and the sum of the first $n$ terms of the series is $S_{n}$

(a) Write down an expression in terms of $a$ and $d$ for

(i) $t_{58}$

(ii) $S_{13}$ (2 marks)

Given that $t_{58}=S_{13}$

(b) show that $d=-\displaystyle\frac{4}{7} a$ (2 marks)

(c) show that $t_{176}=S_{21}$ (4 marks)

(d) find the value of $r$ when $t_{r}=5 t_{9}$ (3 marks)

6. (2014/jan/paper01/q1)
Find $\displaystyle\sum_{y=4}^{40}(7 r-2)$ (4 marks)

7. (2014/jan/paper01/q4)
The sum of the first $n$ terms of an arithmetic series is $2 n(n+3)$

Find

(a) the first term of the series, (1 mark)

(b) the common difference of the series, (3 marks)

(c) the 25 th term of the series. (2 marks)

8. (2014/june/paper01/q4)
The $3 \mathrm{rd}$ term of an arithmetic series is 108 and the 12 th term is 54. Find

(a) the common difference of the series, (3 marks)

(b) the first term of the series. (1 mark)

The sum of the first $n$ terms of the series is $S_{n}$

(c) Show that $S_{n}=3 n(41-n)$ (3 marks)

Given that $S_{n}=1200$

(d) find the two possible values of $n$. (4 marks)

9. (2014/june/paper02/q2)
Evaluate $\displaystyle\sum_{r=5}^{60}(2 r+7)$ (4 marks)

10. (2015/jan/paper02/q7)
The first term of an arithmetic series is $-14$ and the common difference is 4

(a) Find the 15 th term of the series. (2 marks)

(b) Find the sum of the first 25 terms of the series. (3 marks)

The sum of nine consecutive terms of the series is 1422

(c) Find the smallest of these nine terms. (5 marks)

11. ( $2015 /$ june/paper01/q4)
The $\operatorname{sum} S_{n}$ of the first $n$ terms of an arithmetic series is given by $S_{n}=2 n(10-n)$

(a) Write down the first term of the series. (1 mark)

(b) Find the common difference of the series. (2 marks)

Given that $S_{n}>-50$

(c) (i) write down an inequality satisfied by $n$,

(ii) hence find the largest value of $n$ for which $S_{n}>-50$ (4 marks)

12. (2015/june/paper02/q1)
(a) Show that $\displaystyle\sum_{r=1}^{n} r=\displaystyle\frac{n}{2}(1+n)$ (2 marks)

(b) Hence find the sum of all the integers from 1 to 100 inclusive that are not multiples of 7 (3 marks)

13. (2016/jan/paper01/q4)
An arithmetic series has first term $p$ and common difference $p$ where $p \neq 0$ A geometric series also has first term $p$. The common ratio of this geometric series is $r$. The sum of the first three terms of the arithmetic series is equal to the sum of the first three terms of the geometric series.

Given that $r>0$

show that $r=\displaystyle\frac{-1+\sqrt{21}}{2}$ (5 marks)

14. (2016/jan/paper02/q8)
The $n$th term of an arithmetic series is $t_{n}$ where $t_{n}=2 n-3$

The sum of the first $n$ terms of the series is $S_{n}$

(a) Show that $S_{n}=n(n-2)$ (4 marks)

(b) Find the value of $n$ such that $5 t_{n+2}=3 S_{n-3}$ (5 marks)

15. (2016/june/paper01/q4)
The $n$th term of an arithmetic series is $t_{n}$ and the sum of the first $n$ terms of the series is $S_{n}$ Given that $S_{2}=\displaystyle\frac{2}{3} t_{5}$ and that $S_{4}=t_{10}+3$

(a) find

(i) the common difference of the series,

(ii) the first term of the series. (5 marks)

Given also that $S_{p+2}-S_{p}=110$

(b) find the value of $p$ (3 marks)

16. (2017/jan/paper02/q6)
The sum of the first 21 terms of an arithmetic series is 987 and the 8 th term of the series is 35

The first term of the series is $a$ and the common difference is $d$.

(a) Find the value of

(i) $a$,

(ii) $d .$ (5 marks)

The sum, $S_{n}$, of the first $n$ terms of the series is given by $S_{n}=\displaystyle\sum_{r=1}^{n}(A r+B)$, where $A$ and $B$ are integers.

(b) Find the value of

(i) $A$,

(ii) $B$. (3 marks)

(c) Find the least value of $n$ such that $S_{n}>2000$ (5 marks)

17. (2017/june/paper02/q11)
(a) Show that $\log p q^{4}-\log p q^{2}=\log p q^{6}-\log p q^{4}$ (3 marks)

Given that $\log p q^{2}$ and $\log p q^{4}$ are the second and third terms of an arithmetic series, find

(b) the first term of the series, (3 marks)

(c) the sum of the first $n$ terms of the series.

Give your answer in the form $n \log p q^{s}$, expressing $s$ in terms of $n$. (4 marks)

18. (2018/jan/paper02/q2)
(a) Show that $\displaystyle\sum_{r=1}^{n}(3 r+2)=\displaystyle\frac{n}{2}(7+3 n)$ (2 marks)

(b) Hence, or otherwise, evaluate $\displaystyle\sum_{r=10}^{20}(3 r+2)$ (3 marks)

19. (2018/june/paper01/q9)
The 4 th term of an arithmetic series is 108 and the 11 th term is 80

Find

(a) (i) the common difference of the series,

(ii) the first term of the series. (4 marks)

The sum of the first $n$ terms of the series is $S_{n}$

(b) Show that $S_{n}=2 n(61-n)$ (3 marks)

Given that $S_{n}=1100$

(c) find the two possible values of $n$. (4 marks)

20. (2019/juneR $/$ paper01/q6)
(a) Show that $\displaystyle\sum_{r=1}^{n}(4 r-3)=n(2 n-1)$ (3 marks)

(b) Hence, or otherwise, find the least value of $n$ such that $\displaystyle\sum_{r=1}^{n}(4 r-3)>1000$ (3 marks)

Given that $S_{n}=n(2 n-1), t_{n}=(4 n-3)$ and that $18+3 t_{n+7}=S_{n+4}$

(c) find the value of $n$. (4 marks)

21. (2011/june/paper01/q6)
The third term of an arithmetic series is 70 and the sum of the first 10 terms of the series is 450

(a) Calculate the common difference of the series. (4 marks)

The sum of the first $n$ terms of the series is $S_{n}$

Given that $S_{n} \geqslant 350$

(b) find the set of possible values of $n$ (6 marks)

22. (2019/june/paper02/q7)
The sum of the first $n$ terms of an arithmetic series is $A_{n}$ where

$$A_{n}=\displaystyle\sum_{r=1}^{n}(4 r+5)$$

(a) For this arithmetic series, find

(i) the first term,

(ii) the common difference. (2 marks)

The sum of the first $n$ terms of a geometric series is $G_{n}$ where

$$G_{n}=\displaystyle\sum_{r=1}^{n} 4(3)^{r-1}$$

(b) For this geometric series, find

(i) the first term,

(ii) the common ratio. (2 marks)

(c) Find the value of $n$ for which $A_{14}-6=G_{n}$ (5 marks)

1. 345

2.(a) $t_{r}=\log _{a} 2^{r}$ (b) $\log _{a} 2$ (c) Show (d) $n \log _{a} 3$

3.(a) show (b) (c) $n=12$

4.(a) Show (b) $d=4$ (c) $u_{12}=49$ (d) $p=9$

5.(a)(i) $t_{5_{8}}=a+57 d$ (ii) $S_{13}=\displaystyle\frac{13}{2}(2 a+12 d)$ (b) Show (c) Show (d) $r=34$

6. 5624

7.(a) $a=8$ (b) $d=4 \quad$ (c) $u_{25}=104$

8.(a) $\quad d=-6$ (b) Show (c) $n=16,25$

9. 4032

10.(a) $\quad u_{15}=42$ (b) $S_{25}=850$ (c) $a=142$

11.(a) $a=18$ (b) $d=-4$ (c) (i) $2 n(10-n)>-50$ (ii) $n=12$

12.(a) Show (b) 4315

13.Show

14.(a) Show (b) $n=10$

15.(a)(i) $d=4$ (ii) $a=5$ (b) $p=12$

16.$(a)(i) a=7($ ii) $d=4 \quad(b)(i) A=4 \quad$ (ii) $B=3$ (c) $n=31$

17.(a) Show (b) $a=\log p$ (c) $S_{n}=n \log \left(p q^{(n-1)}\right)$

18.(a) Show (b) 517

19.(a) (i) $d=-4$ (ii) $a=120$ (b) Show (c) $n=11,50$

20.(a) Show (b) $n=23$ (c) $n=5$

21.(a) $d=-10$ (b) $n=5,6, \ldots, 14$

22.$(a)(i) \quad a=9$ (ii) $d=4$ (b) $(i) a=4$ (ii) $r=3$ (c) $n=5$