# FPM Geometric Series (Chapter 5)

$\def\frac{\dfrac}$

1. (2011/june/Paper02/q8)
The sum of the first and third terms of a geometric series is 100

The sum of the second and third terms is 60

(a) Find the two possible values of the common ratio of the series. (5 marks)

Given that the series is convergent, find

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

(c) the least number of terms for which the sum is greater than $159.9$ (4 marks)

2. (2012/jan/paper02/q10)
The sm of the first and third terms of a geometric series $G$ is 104

The sum of the second and third terms of $G$ is 24

Given that $G$ is convergent and that the sum to infinity is $S$, find

(a) the common ratio of $G$ (4 marks)

(b) the value of $S$ (4 marks)

The sum of the first and third terms of another geometric series $H$ is also 104 and the sum of the second and third terms of $H$ is 24

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

(c) Write down the common ratio of $H$ (1 mark)

(d) Find the least value of $n$ for which $S_{n}>S$ (6 marks)

3. $(2012 /$ june $/$ paper02 $/ \mathrm{q} 6$ )
The first term of a geometric series $S$ is $\sqrt{2}$

The second term of $S$ is $\sqrt{2}-2$

(a) (i) Find the exact value of the common ratio of $S$.

(ii) Find the third term of $S$, giving your answer in the form $a \sqrt{2}+b$, where $a$ and $b$ are integers. (5 marks)

(b) (i) Explain why the series is convergent.

(ii) Find the sum to infinity of $S$. (3 marks)

4. (2013/jan/paper02/q9)
The third and fifth terms of a geometric series $S$ are 48 and 768 respectively. Find

(a) the two possible values of the common ratio of $S$, (3 marks)

(b) the first term of $S$. (1 mark)

Given that the sum of the first 5 terms of $S$ is 615

(c) find the sum of the first 9 terms of $S$. (4 marks)

Another geometric series $T$ has the same first term as $S$. The common ratio of $T$ is $\dfrac{1}{r}$ where $r$ is one of the values obtained in part (a). The $n$th term of $T$ is $t_{n}$

Given that $t_{2}>t_{3}$

(d) find the common ratio of $T$. (1 marks)

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

(e) Writing down all the numbers on your calculator display, find $T_{9}$ (2 marks)

The sum to infinity of $T$ is $T_{\infty}$

Given that $T_{\infty}-T_{n}>0.002$

(f) find the greatest value of $n$. (5 marks)

5. ( $2013 /$ june $/$ paper02 $/ \mathrm{q} 4$ )
The $n$th term of a geometric series is $t_{n}$ and the common ratio is $r$, where $r>0$ Given that $t_{1}=1$

(a) write down an expression in terms of $r$ and $n$ for $t_{n}$ (1 marks)

Given also that $t_{n}+t_{n+1}=t_{n+2}$

(b) show that $r=\dfrac{1+\sqrt{5}}{2}$ (4 marks)

(c) find the exact value of $t_{4}$ giving your answer in the form $f+g \sqrt{h}$, where $f, g$ and $h$ are integers. (3 marks)

6. (2014/jan/paper02/q 10$)$
The sum of the second and third terms of a convergent geometric series is $7.5$

The sum to infinity, $S$, of the series is 20

The common ratio of the series is $r$.

(a) Show that $r$ is a root of the equation

$$8 r^{3}-8 r+3=0$$ (4 marks)

(b) Show that $r=\dfrac{1}{2}$ is a root of this equation. (1 mark)

Given that $r < 0.6$

(c) show that $\dfrac{1}{2}$ is the only possible value of $r$. (4 marks)

(d) Find the first term of the series. (2 marks)

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

(e) Find the least value of $n$ for which $S_{n}$ exceeds $99 \%$ of $S$. (6 marks)

7. (2014/june/paper02/q6)
The sum to infinity of a convergent geometric series with common ratio $r$ is $S$.

Given that $S=200$ and that the sum of the first 3 terms is 175

(a) find the value of $r$, (4 marks)

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

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

Given also that $\dfrac{S_{n}}{S}=\dfrac{255}{256}$

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

8. ( $2015 /$ june/paper02/q3)
Every term of a convergent geometric series is positive. The difference between the third term and the fourth term is twice the fifth term.

(a) Show that the common ratio of the series is $\dfrac{1}{2}$ (3 marks)

The sum to infinity of this convergent series is 400

Find

(b) the first term of the series, (2 marks)

(c) the sum of the first 10 terms of the series, writing down all the digits on your calculator display. (2 marks)

9. ( $2016 /$ june/paper02/q3)
A geometric series has first term $(11 x-3)$, second term $(5 x+3)$ and third term $(3 x-3)$.

(a) Find the two possible values of $x$. (4 marks)

For each of your values of $x$

(b) find the corresponding value of the common ratio of the series. (3 marks)

Given that the series is convergent,

(c) find the sum to infinity of the series. (3 marks)

10. (2017/jan/paper01/q4)
The $n$th term of a geometric series is $t_{n}$ and the common ratio is $r$.

Given that $t_{2}+t_{5}=\dfrac{28}{81}$ and $t_{2}-t_{5}=\dfrac{76}{405}$

(a) (i) show that $r=\dfrac{2}{3}$

(ii) find the first term of the series. (6 marks)

(b) Find the sum to infinity of this geometric series. (2 marks)

11. (2017/june/paper01/q6)
The sum of the first term and the third term of a geometric series is 250

The sum of the second term and the third term of the series is 150

The common ratio of the series is $r$.

(a) Find the two possible values of $r$. (5 marks)

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

Given that $r>0$ and that $S_{n}>399.99$

(b) find the least value of $n$. (6 marks)

12. $(2018 / \mathrm{jan} /$ paper01/q8)
The sixth term of a geometric series $G$, with common ratio $r(r \neq 0)$, is four times the second term.

(a) Find the two possible exact values of $r$. (2 marks)

The sum of the third and seventh terms of $G$ is 30

(b) Find the first term of the series. (3 marks)

Given that $r>0$

(c) find the sum of the first 10 terms of $G$ (2 marks)

Given that $t_{n}$ is the $n$th term of $G$,

(d) find the least value of $n$ for which $t_{n}>2400$ (3 marks)

13. (2018/june/рареr02/q5)
The sum of the first term and the third term of a geometric series is 75

The sum of the second term and the third term is 45

(a) Find the two possible values of the common ratio of the series. (5 marks)

Given that the series is convergent with sum to infinity $S$,

(b) find the value of $S$. (3 marks)

14. (2019/juneR/paper02/q7)
The $n$th term of a geometric series $G$ is $u_{n}$

The first term of $G$ is $a$ and the common ratio of $G$ is $r$, where $r>0$

Given that $u_{3}=4$ and that $u_{7}=16$

(a) (i) show that $r=\sqrt{2}$

(ii) find the value of $a$. (3 marks)

(b) Find the least value of $n$ for which $u_{n}>500$ (4 marks)

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

(c) Find $S_{20}$ Give your answer in the form $p(1+\sqrt{2})$ where $p$ is an integer. (4 marks)

1.(a) $r=\dfrac{1}{2},-3$ (b) $a=80$ (c) $n=11$

2.(a) $r=\dfrac{1}{5}$ (b) $S=125$ (c) $-\dfrac{3}{2}$ (d) $n=7$

3.(a) (i) $r=1-\sqrt{2}$ (ii) $t_{3}=3 \sqrt{2}-4$ (b) (i) $(r \mid < 1$ (ii) 1

4.(a) $r=\pm 4$ (b) $a=3$ (c) $S_{q}=157287$ (d) $r=\dfrac{1}{4}$ (e) $T_{9}=3.999984741$ (f) $n=5$

5.(a) $t_{n}=r^{n-1}$ (b) show (c) $t_{4}=2+\sqrt{5}$

6.(a) Show (b) show (c) show $(d) a=10$ (e) $n=7$

7.(a) $\quad r=\dfrac{1}{2}$ (b) $a=100$ (c) $n=8$

8.(a) Show (b) $a=200$ (c) $399 \dfrac{39}{64}$

9.(a) $\quad x=0,9$ (b) $x=0, r=-1 ; x=9, r=\dfrac{1}{2}$ (c) $S_{\infty}=192$

10.(a)(i) Show (ii) $a=\dfrac{2}{5}$(b) $S=\dfrac{6}{5}$

11.(a) $r=\dfrac{1}{2},-3$ (b) $n=16$

12.(a) $r=\pm \sqrt{2}$ (b) $a=3$ (c) $S_{10}= 93(\sqrt{2}+1)$ (d) $n=21$

13.(a) $r=-3,\dfrac{1}{2}$ (b)$S=120$

14.(a) (i) Show (ii) $a=2$ (b) $n=17$ (c) $S_{20}=2046(1+\sqrt 2)$