Sequences and Series

Maths, HSSC-KPK First Year. This chapter is split into 44 topics and has 1,087 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Sequence versus series 20 questions
2. Geometric mean between two numbers 24 questions
3. A.P. word problems: sums 26 questions
4. Summation notation 26 questions
5. A.M. and G.M. relations 24 questions
6. Arithmetic sequence: definition and common difference 25 questions
7. Numbers in A.P. from sums and sums of powers 22 questions
8. Writing the nth term from a pattern 22 questions
9. Sums of multiples and integers in a range 26 questions
10. Inserting n harmonic means 25 questions
11. Finding missing components of an A.P. 26 questions
12. Sum of the first n terms of a G.P. 27 questions
13. Missing components of a G.P. and its sum 26 questions
14. When does an infinite G.P. have a sum 24 questions
15. Number of terms and conditions for a G.P. 25 questions
16. The nth term of an A.P. 27 questions
17. Terms of a sequence from its general term 26 questions
18. Number of terms and testing for an A.P. 25 questions
19. Review Exercise 4 problems 25 questions
20. Sequence: definition, terms and introduction 26 questions
21. Recurring decimals as fractions 26 questions
22. Finite and infinite sequences 22 questions
23. Sum of an A.P. from a, d and n 27 questions
24. Geometric sequence: definition and common ratio 25 questions
25. Special sequences: triangle, factorial and Fibonacci 24 questions
26. G.P. word problems 26 questions
27. Arithmetic series and the sum formula 22 questions
28. Recursively defined sequences 25 questions
29. Three terms in A.P. and finding unknown constants 22 questions
30. A.P. from the sum S(n) and advanced problems 25 questions
31. Harmonic sequence: definition 22 questions
32. Arithmetic mean between two numbers 25 questions
33. Inserting n arithmetic means 27 questions
34. The nth term of a G.P. 26 questions
35. Geometric series and the sum formula 22 questions
36. A.P. word problems: terms 26 questions
37. Finding the A.P. from two given terms 26 questions
38. The nth term of an H.P. 24 questions
39. Pascal sequences and Pascal's triangle 24 questions
40. Relation among A.M., G.M. and H.M. 22 questions
41. Inserting n geometric means 25 questions
42. Harmonic mean between two numbers 25 questions
43. Finding the G.P. from two given terms 26 questions
44. Sum of an infinite geometric series 26 questions

Sample questions: Sequences and Series

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Exercise 4.1 Q1(iii): the sequence ..., −4, 0, 4, 8, ..., 60 has a last term 60 but no first term, so it is

• A. a geometric sequence
• B. infinite(correct answer)
• C. finite
• D. finite because it has a last term

Answer: B.A sequence is finite only if it has both a first and a last term; this one has no first term, so it is infinite (the book's key agrees).

Q2. Which of the following numbers is a term of the A.P. 9, 16, 23, …?

• A. 128(correct answer)
• B. 129
• C. 132
• D. 127

Answer: A.A term has the form 9 + 7(n − 1); 128 gives n = 18, whereas the other numbers leave a non-zero remainder when (x − 9) is divided by 7.

Q3. Which of the following numbers is a term of the A.P. 1, 8, 15, …?

• A. 155(correct answer)
• B. 156
• C. 154
• D. 159

Answer: A.A term has the form 1 + 7(n − 1); 155 gives n = 23, whereas the other numbers leave a non-zero remainder when (x − 1) is divided by 7.

Q4. Exercise 4.2 Q15: the value of n for which (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the arithmetic mean of a and b is

• A. 0(correct answer)
• B. 1/2
• C. −1
• D. 1

Answer: A.Putting n = 0 gives (a + b)/2, the arithmetic mean. (Cross-multiplying gives (a − b)(aⁿ − bⁿ) = 0, so n = 0 for a ≠ b; the book's key shows 0.)

Q5. Exercise 4.2 Q11: climbing 1000 feet in the first hour and 100 feet more in each succeeding hour, they reach the top of a 5400 feet hill after how many hours?

• A. 44
• B. 54
• C. 46
• D. 45(correct answer)

Answer: D.The distance per hour is 1000, 1100, ...; the book takes the hourly gain as 1000 + 100(n − 1) = 5400, so n = 45 (the book's key says 45 hours).

Q6. Exercise 4.3 Q7: the sum of 1 + 3 − 5 + 7 + 9 − 11 + 13 + 15 − 17 + ... up to 3n terms is

• A. 3n(n − 1)
• B. n(3n − 2)
• C. n(3n + 4)
• D. n(3n − 4)(correct answer)

Answer: D.Each group of three terms gives −1, 5, 11, ..., an A.P. with first term −1 and d = 6, so the sum is (n/2)(−2 + 6(n − 1)) = n(3n − 4) (the book's key agrees).

Q7. Exercise 4.5 Q1(ii): 8 + 4 + 2 + 1 + ... + 1/16 equals

• A. 127/8
• B. 31/2
• C. 255/16(correct answer)
• D. 255/8

Answer: C.a₁ = 8, r = 1/2, last term 1/16 (n = 8): S = (8 − (1/16)(1/2))/(1/2) = 255/16 = 15.9375. (The book's key prints 255/7, which is not equal to the sum.)

Q8. The sum of an infinite G.P. is 12 and the sum of the squares of its terms is 432/5. The common ratio is

• A. 1/4(correct answer)
• B. −1/4
• C. 3/4
• D. 1/16

Answer: A.Dividing the sum of squares by the sum gives a/(1 + r) = 36/5; combining with a/(1 − r) = 12 gives (1 + r)/(1 − r) = 5/3, so r = 1/4.

Q9. Review Exercise 4 Q9: three numbers in G.P. have sum 26 and product 216. The numbers are

• A. 1, 6, 36
• B. 2, 6, 18(correct answer)
• C. 3, 6, 12
• D. 4, 6, 9

Answer: B.The middle number is the cube root of 216, i.e. 6, and 6/r + 6 + 6r = 26 gives r = 3 (or 1/3); the numbers are 2, 6, 18 (the book's key agrees).

Q10. Review Exercise 4 Q11: a balloon has risen 100 ft and in each succeeding minute rises half as far as in the previous minute. Its total height after the 5 further minutes is

• A. 187.5 ft
• B. 196.875 ft(correct answer)
• C. 200 ft
• D. 193.75 ft

Answer: B.The five further rises 50, 25, 12.5, 6.25, 3.125 add to 96.875; with the first 100 ft the height is 196.875 ft (the book's key agrees). The question wording is ambiguous.