# AP and GP Series (IB Standard Level)

$\def\D{\displaystyle} \def\iixi#1#2{\D\left(\begin{array}{c} #1\\#2 \end{array}\right)}$

1.) In an arithmetic sequence, $\D u_1 = 2$ and $\D u_3 = 8.$
(a) Find $\D d.$
(b) Find $\D u_{20}.$
(c) Find $\D S_{20}.$ (Total 6 marks)

2.) In an arithmetic sequence $\D u_1 = 7, u_{20} = 64$ and $\D u_n = 3709.$
(a) Find the value of the common difference.
(b) Find the value of $\D n.$ (Total 5 marks)

3.) Consider the arithmetic sequence 3, 9, 15, ..., 1353.
(a) Write down the common difference.
(b) Find the number of terms in the sequence.
(c) Find the sum of the sequence. (Total 6 marks)

4.) An arithmetic sequence, $\D u_1, u_2, u_3, \ldots ,$ has $\D d = 11$ and $\D u_{27} = 263.$
(a) Find $u_1.$
(b) (i) Given that $\D u_n = 516,$ find the value of $\D n.$
(ii) For this value of $\D n,$ find $\D S_n.$ (Total 6 marks)

5.) The first three terms of an infinite geometric sequence are 32, 16 and 8.
(a) Write down the value of $\D r.$
(b) Find $\D u_6.$
(c) Find the sum to infinity of this sequence. (Total 5 marks)

6.) The $\D n^{th}$ term of an arithmetic sequence is given by $\D u_n = 5 + 2n.$
(a) Write down the common difference.
(b) (i) Given that the $\D n^{th}$ term of this sequence is 115, find the value of $\D n.$
(ii) For this value of $\D n,$ find the sum of the sequence. (Total 6 marks)

7.) In an arithmetic series, the first term is –7 and the sum of the first 20 terms is 620.
(a) Find the common difference.
(b) Find the value of the $\D 78 ^{th}$ term. (Total 5 marks)

8.) In a geometric series, $\D u1 = \frac{1}{81}$ and $\D u_4 = \frac{1}{3} .$
(a) Find the value of $\D r.$
(b) Find the smallest value of $\D n$ for which $\D S_n > 40.$ (Total 7 marks)

9.) (a) Expand $\D \sum_{r=4}^{7} 2^r$ as the sum of four terms.
(b) (i) Find the value of $\D \sum_{r=4}^{30} 2^r.$
(ii) Explain why $\D \sum_{r=4}^{\infty } 2^r$ cannot be evaluated.  (Total 7 marks)

10.) In an arithmetic sequence, $\D S_{40} = 1900$ and $\D u_{40} = 106.$ Find the value of $\D u_1$ and of $\D d.$ (Total 6 marks)

11.) Consider the arithmetic sequence 2, 5, 8, 11, ....
(a) Find $\D u_{101}.$
(b) Find the value of $\D n$ so that $\D u_n = 152.$ (Total 6 marks)

12.) Consider the infinite geometric sequence 3000, - 1800, 1080, -648, … .
(a) Find the common ratio.
(b) Find the 10 th term.
(c) Find the exact sum of the infinite sequence. (Total 6 marks)

13.) Consider the infinite geometric sequence $\D 3, 3(0.9), 3(0.9)^2, 3(0.9)^3, … .$
(a) Write down the 10 th term of the sequence. Do not simplify your answer.
(b) Find the sum of the infinite sequence. (Total 5 marks)

14.) In an arithmetic sequence $\D u_{21} = -37$ and $\D u_4 = -3.$
(a) Find (i) the common difference; (ii) the first term.
(b) Find $\D S_{10}.$ (Total 7 marks)

15.) Let $\D u_n = 3 - 2n.$
(a) Write down the value of $\D u_1, u_2,$ and $\D u_3.$
(b) Find $\D \sum_{n=1}^{20} (3-2n)$ (Total 6 marks)

16.) A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row.
(a) Calculate the number of seats in the 20th row.
(b) Calculate the total number of seats. (Total 6 marks)

17.) A sum of \$5000 is invested at a compound interest rate of 6.3 % per annum. (a) Write down an expression for the value of the investment after n full years. (b) What will be the value of the investment at the end of five years? (c) The value of the investment will exceed \$ 10 000 after n full years.
(i) Write down an inequality to represent this information.
(ii) Calculate the minimum value of n. (Total 6 marks)

18.) Consider the infinite geometric sequence 25, 5, 1, 0.2, … .
(a) Find the common ratio.
(b) Find (i) the 10 th term; (ii) an expression for the n th term.
(c) Find the sum of the infinite sequence. (Total 6 marks)

19.) The first four terms of a sequence are 18, 54, 162, 486.
(a) Use all four terms to show that this is a geometric sequence.
(b) (i) Find an expression for the n th term of this geometric sequence.
(ii) If the n th term of the sequence is 1062 882, find the value of n. (Total 6 marks)

20.) (a) Write down the first three terms of the sequence $\D u_n = 3n,$ for $\D n\ge 1.$
(b) Find (i) $\D \sum_{n=1}^{20} 3n$
(ii) $\D \sum_{n=21}^{100} 3n$  (Total 6 marks)

21.) Consider the infinite geometric series 405 + 270 + 180 +....
(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the fifteenth term of this series.
(c) Find the exact value of the sum of the infinite series. (Total 6 marks)

22.) (a) Consider the geometric sequence -3, 6, -12, 24, ….
(i) Write down the common ratio.
(ii) Find the 15 th term. Consider the sequence $\D x - 3, x +1, 2x + 8,\ldots.$
(b) When $\D x = 5,$ the sequence is geometric.
(i) Write down the first three terms.
(ii) Find the common ratio.
(c) Find the other value of $\D x$ for which the sequence is geometric.
(d) For this value of $\D x,$ find
(i) the common ratio;
(ii) the sum of the infinite sequence. (Total 12 marks)

1.
3,59 610
2. 3,1235
3. 6,226,153228
4. -23,50,12325
5. 1/2,1,64
6. 2,55,3355
7. 4,301
8. 3,8
9.  240,2147483632
10. -11,3
11. 302,51
12. -0.6,-30.2,1875
13. 1.162,30
14. -2,3,-60
15. 1,-1,-3,-360
16. 53,680
17. 5000(1.063^n),6786.35,...,12
18. 0.2,1/78125,...,31.25
19. r=3,18x3^(n-1),11
20. 3,6,9:630,14520
21. 2/3,1.39,1215
22. -2,-49152;2,6,28;3,-5,-16