GP Series (Edexcel IGCSE, Further Pure Mathematics)

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1. (Edexcel Further Pure  Mathematics 2011, june, paper 2, no.8)
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)
Given that the series is convergent, find
(b) the first term of the series, (2)
(c) the least number of terms for which the sum is greater than 159.9 (4)

2.  (Edexcel Further Pure  Mathematics, 2012, jan, Paper 2, no 10)
The sum of the first and third terms of a geometric series $\D G$ is 104. The sum of the second and third terms of $\D G$ is 24 Given that $\D G$ is convergent and that the sum to infinity is $\D S,$ find
(a) the common ratio of $\D G.$ (4)
(b) the value of $\D S.$ (4)
The sum of the first and third terms of another geometric series $\D H$ is also 104 and the sum of the second and third terms of $\D H$ is 24. The sum of the first n terms of H is $\D S_n.$
(c) Write down the common ratio of $\D H$. (1)
(d) Find the least value of $\D n$ for which $\D S_n> S.$  (6)

3. (Edexcel Further Pure Mathematics, 2012, june, Paper 2, no 6)
The first term of a geometric series $\D S$ is $\D \sqrt{2}.$ The second term of $\D S$ is $\D \sqrt{2} − 2.$
(a) (i) Find the exact value of the common ratio of $\D S.$
(ii) Find the third term of $\D S,$ giving your answer in the form $\D a \sqrt{2} + b,$ where $\D a$ and $\D b$ are integers. (5)
(b) (i) Explain why the series is convergent.
(ii) Find the sum to infinity of $\D S.$ (3)

4. (Edexcel Further Pure  Mathematics, 2013, Jan, Paper 2, no 9)
The third and fifth terms of a geometric series $\D S$ are 48 and 768 respectively. Find
(a) the two possible values of the common ratio of $\D S,$ (3)
(b) the first term of $\D S.$ (1)
Given that the sum of the first 5 terms of $\D S$ is 615.
(c) find the sum of the first 9 terms of $\D S.$ (4)
Another geometric series $\D T$ has the same first term as $\D S.$ The common ratio of $\D T$ is $\D \frac{1}{r},$ where $\D r$ is one of the values obtained in part (a). The $\D n^{th}$ term of $\D T$ is $\D t_n.$ Given that $\D t_2 > t_3.$
(d) find the common ratio of $\D T.$ (1)
The sum of the first $\D n$ terms of $\D T$ is $\D T_n.$
(e) Writing down all the numbers on your calculator display, find $\D T_9.$ (2)
The sum to infinity of $\D T$ is $\D T_{\infty}.$ Given that $\D T_{\infty} - T_n > 0.002.$
(f) find the greatest value of $\D n.$ (5)

5. (Edexcel Further Pure  Mathematics, 2013, june, Paper 2, no 4)
The $\D n_{th}$ term of a geometric series is $\D t_n$ and the common ratio is $\D r,$ where $\D r > 0.$ Given that $\D t_1 = 1.$
(a) write down an expression in terms of $\D r$ and $\D n$ for $\D t_n.$ (1)
Given also that $\D t_n + t_{n+1} = t_{n+2},$
(b) show that $\D r = \frac{1+\sqrt{5}}{2}$ (4)
(c) find the exact value of $\D t_4$ giving your answer in the form $\D f + g\sqrt{h},$ where $\D f, g$ and $\D h$ are integers. (3)

6. (Edexcel Further Pure  Mathematics, 2014, Jan, Paper 2, no 10)
The sum of the second and third terms of a convergent geometric series is 7.5. The sum to infinity, $\D S,$ of the series is 20. The common ratio of the series is $\D r.$
(a) Show that $\D r$ is a root of the equation $\D 8r^3 – 8r + 3 = 0.$ (4)
(b) Show that $\D r =\frac{1}{2}$ is a root of this equation. (1)
Given that $\D r < 0.6.$
(c) show that $\frac{1}{2}$ is the only possible value of $\D r.$ (4)
(d) Find the first term of the series. (2)
The sum of the first $\D n$ terms of the series is $\D S_n$.
(e) Find the least value of $\D n$ for which Sn exceeds 99\% of $\D S.$ (6)

7.(Edexcel Further Pure  Mathematics, 2015, june, Paper 2, No 3)
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 $\D \frac{1}{2}.$ (3)
The sum to infinity of this convergent series is 400. Find
(b) the first term of the series, (2)
(c) the sum of the first 10 terms of the series, writing down all the digits on your calculator display. (2)

1.(a) $\D r=1/2,-3$
(b) $\D a=80$
(c) $\D n=11$
2. (a) $\D r=1/5 (r=-3/2)$
(b) $\D a=100,S=125$
(c) $r'=-3/2$
(d) $\D n=7$
3. (a)(i) $\D r=1-\sqrt{2}$
(ii) $\D 3\sqrt{2}-4$
(b)(i) $\D |r|<1$
(ii) $\D S=1$
4. (a) $\D \pm 4$
(b) $\D a=3$
(c) $\D r=-4, S_9=157287$
(d) $\D r=\frac{1}{4}$
(e) $\D T_9=3.999984741$
(f) $\D n=5$
5. (a) $\D t_n=r^{n-1}$
(b)
(c) $\D 2+\sqrt{5}$
6. (a)(b)(c)
(d) $\D a=10$
(e) $\D n=7$
7. (a)
(b) $\D a=200$
(c) $S_{10}=399.609375$