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INTERNATIONAL BACCALAUREATE
Mathematics: analysis and approaches
Math AA
EXERCISES [Math-AA 1.2-1.3] ARITHMETIC SEQUENCES Compiled by Christos Nikolaidis
O. Practice questions SEQUENCES (IN GENERAL)
1. [Maximum mark: 6] [without GDC] A sequence may be given by the first few terms. (a) Complete the following table.
Sequence u n : 1, 2, 3, 4, 5, 6, ….
20-th term
u n in terms of n.
S 1
S 2
S 3
[3]
(b) Complete the following table.
Sequence u n : 3, 4, 5, 6, 7, 8, ….
20-th term
u n in terms of^ n.
S 1
S 2
S 3
[3]
2. [Maximum mark: 12] [without GDC]
A sequence may be given by a general formula for un.
(a) Complete the following table.
Sequence un : un 2 n
First three terms
u 10
S 1
S 2
S 3
[4]
(b) Complete the following table.
Sequence un : u n n^2
First three terms
u 10
S 1
S 2
S 3
[4]
(c) Complete the following table.
Sequence un : u n 2 n
First three terms
u 10
S 1
S 2
S 3
[4]
REMARK for question 2:
Use your GDC (recursion, TYPE F1: an in terms of n ), to obtain all the results above.
4. [Maximum mark: 8] [without GDC] A series may be given by Sigma notation. Express each of the following series as a sum of three terms and find the result. 3
r 1 (3 )^ r
(^3 )
r 1^ r
3
r
r
(^3 )
r 1 (3^^ r^ 1)
REMARK for question 4: Use your GDC (Run-matrix – Math – Sigma notation), to obtain all the results above.
5. [Maximum mark: 4] [with GDC] (a) Find (^10 )
A r 1 (2 r 1)
(^20 )
B r 1 (2 r 1)
(^20 )
C r 11 (2 r 1)
. [3]
(b) Write down a relation between A , B and C. [1]
ARITHMETIC SEQUENCES
6. [Maximum mark: 10] [with / without GDC] Consider the arithmetic sequence 11, 15, 19, 23, …
(a) Find u 101 and S 101 [4]
(b) Find the sum of the first 20 terms. [2]
(c) Express the general term un in the form un an b. [2]
(d) Hence find the value of n given that u n 51. [2]
8. [Maximum mark: 6] [with / without GDC] The 5th^ term of an arithmetic sequence is 30, while the 13th^ term is 70.
(a) Find the first term u 1 and the common difference d. [3]
(b) Find the general term un in terms of n. [1]
(c) Find the 20th^ term of the sequence. [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. ..................................................................................................................................
9. [Maximum mark: 5] [without GDC]
In an arithmetic sequence, S 1 10 and S 2 25.
(a) Write down u 1 and u 2. [2]
(b) Find the common difference d. [1]
(c) Find S 3 and S 4. [2]
10. [Maximum mark: 7] [with GDC]
In an arithmetic sequence, S 5 75 and S 12 348.
(a) Write down two simultaneous equations in u 1 and d. [3]
(b) Hence find the first term u 1 and the common difference d. [2]
(c) Find S 32. [2]
11. [Maximum mark: 7] [with GDC]
In an arithmetic sequence, let u 5 48 , and S 10 515
(a) Write down two simultaneous equations in u 1 and d. [3]
(b) Hence find the first term u 1 and the common difference d. [2]
(c) Find u 47. [2]
14* [Maximum mark: 8] [without GDC] Calculate the following sums
(a) (i) r ^31 ( 2 r 1 ) (ii) ^200 r 1 ( 2 r 1 ) (iii) r^200 4 ( 2 r 1 ) [5]
(b) r ^200101 ( 2 r 1 ) [3]
REMARK for question 14: Use your GDC (Run-matrix – Math – Sigma notation), to obtain all the results above.
15.* [Maximum mark: 10] [with GDC] Consider the arithmetic sequence 11, 15, 19, 23, … (a) Find the number of terms which are less than 100. [3] (b) Find the last term which is less than 100. [2] (c) Find the sum of all terms which are less than 100. [2] (d) Find
(i) the greatest value of n such that Sn 1000
(ii) the least value of n such that S n 1000 [3]
18.* [Maximum mark: 8] [without GDC]
The sum of the first n terms of a sequence is given by
S n 4 n^2 n , where^ n^ ^ +.
(a) Find S 4 , S 5 and hence u 5. [3]
(b) Find an expression for un , the n th^ term of the sequence. [3]
(c) Show that the sequence is arithmetic by considering the difference u (^) n un 1_._ [2] ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
19.** [Maximum mark: 15] [with GDC] The positive multiples of 8 form an arithmetic sequence: 8, 16, 24, 32, …
You may find any sum of multiples of 8 by using the sigma notation x^ n 1 8 x .
Find (a) the sum of all multiples of 8 between 1 and 900. [3] (b) the sum of all multiples of 8 between 100 and 900. [3] (c) the sum of all numbers which are multiples of both 8 and 6 , between 1 and 900. [3] (d) the sum of all numbers which are multiples of 8 but not of 6 , between 1 and 900. [3] (e) the sum of all numbers which are multiples of 8 or of 6 , between 1 and 900. [3] ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
22. [Maximum mark: 6] [without GDC]
In an arithmetic sequence, u 1 2 and u 3 8.
(a) Find d. [2]
(b) Find u 20. [2]
(c) Find S 20. [2]
23. [Maximum mark: 4] [without GDC] In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. ..................................................................................................................................
24. [Maximum mark: 6] [with GDC] Arturo goes swimming every week. He swims 200 metres in the first week. Each week he swims 30 m more than the previous week. He continues for one year (52 weeks). (a) How far does Arturo swim in the final week? [3] (b) How far does he swim altogether? [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 25. [Maximum mark: 6] [with GDC] 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. [4] (b) Calculate the total number of seats. [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. ..................................................................................................................................
28. [Maximum mark: 6] [with GDC] A teacher earns an annual salary of 45 000 USD for the first year of her employment Her annual salary increases by 1750 USD each year. (a) Calculate the annual salary for the fifth year of her employment. [3] She remains in this employment for 10 years. (b) Calculate the total salary she earns in this employment during these 10 years. [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 29. [Maximum mark: 4] [with GDC] A tree begins losing its leaves in October. The number of leaves that the tree loses each day increases by the same number on each successive day. Date in October 1 2 3 4 ..................... Number of leaves lost 24 40 56 72 ..................... (a) Calculate the number of leaves that the tree loses on the 21st October. [2] (b) Find the total number of leaves that the tree loses in the 31 days of the month of October. [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. ..................................................................................................................................
30. [Maximum mark: 5] [with GDC]
In an arithmetic sequence u 1 7 , u 20 64 and un 3709.
(a) Find the value of the common difference. [3]
(b) Find the value of n. [2]
31. [Maximum mark: 4] [without GDC] The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, u 1 , and the common difference, d. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. ................................................................................................................................. .................................................................................................................................
