SEQUENCE AND SERIES REVISION QUESTIONS

Sequences and Series Questions

1. Find the 20th term of the arithmetic sequence: 7, 12, 17, ...
2. What is the sum of the first 15 terms of the sequence: 2, 5, 8, 11, ...?
3. Find the 10th term of the geometric sequence: 3, 6, 12, ...
4. Find the sum of the first 8 terms of the geometric sequence: 5, 10, 20, ...
5. In an arithmetic sequence, the 5th term is 22 and the 10th term is 47. Find the first term and common difference.
6. Find the sum of all even numbers between 1 and 100.
7. If the sum of 4 consecutive terms of an arithmetic sequence is 70 and the second term is 15, find the terms.
8. Find the 12th term of an arithmetic sequence whose sum of the first 12 terms is 336.
9. Find the sum to infinity of the geometric series: 8 + 4 + 2 + ...
10. Find the sum of the first 10 terms of a geometric sequence where a = 2 and r = -3.
11. The 6th term of a geometric sequence is 243 and the 3rd term is 9. Find the common ratio.
12. Find the sum of the first 20 terms of an arithmetic sequence if the first term is 1 and the 20th term is 39.
13. The sum of 5 terms of a geometric sequence is 121 and the first term is 1. Find the common ratio.
14. If an arithmetic sequence has a common difference of 5, and its 6th term is equal to the 3rd term of a geometric sequence with a = 2 and r = 3, find the first term of the arithmetic sequence.
15. If the 2nd term of an arithmetic sequence is equal to the 1st term of a geometric sequence, and the 5th term of the arithmetic is equal to the 3rd term of the geometric, find the common difference of the arithmetic sequence in terms of the geometric ratio.
16. In a Fibonacci sequence, find the 10th term.
17. If a sequence follows Fibonacci rule and starts with 3, 5, what is the 7th term?
18. Find the sum of the first 6 terms of the Fibonacci sequence starting with 1, 1.
19. The sum of three consecutive Fibonacci numbers is 20. Find the numbers.
20. Which term in the sequence 1, 1, 2, 3, 5, 8, ... is the first to exceed 100?
21. In an arithmetic sequence, the 4th term is 19 and the 9th term is 39. Find the sum of the first 20 terms.
22. The sum to infinity of a geometric series is 6, and the first term is 2. Find the common ratio.
23. The 2nd term of an arithmetic sequence is 9, and the 5th term is 18. Find the 50th term.
24. If the nth term of a sequence is given by Tₙ = 3n + 2, find the sum of the first 30 terms.
25. Find the number of terms in the arithmetic sequence: 4, 9, 14, ..., 99.
26. If the sum of a geometric series is 121 and the first term is 11, with r > 0, find the number of terms if r = 2.
27. The 1st term of an arithmetic sequence is a and the common difference is d. Show that the sum of the first n terms is n/2 [2a + (n−1)d].
28. If the 1st term of an A.P is also the 1st term of a G.P, and the 3rd term of the A.P is the same as the 2nd term of the G.P, find the relation between their common difference and common ratio.
29. In an arithmetic sequence, T₁ = 5 and Tₙ = 95. If the sum of all terms is 1000, find n.
30. Find the geometric mean between 7 and 63.
31. If the first term of an A.P and a G.P is the same, and the 2nd term of the A.P is equal to the 3rd term of the G.P, find the relation between the common difference and common ratio.
32. The 1st and 2nd terms of an A.P. are equal to the 1st and 3rd terms of a G.P. respectively. Find the relation between d and r.
33. If the 2nd term of an A.P equals the 1st term of a G.P, and the 4th term of the A.P equals the 2nd term of the G.P, express d in terms of r.
34. If a is the first term of both an A.P and a G.P, and the 4th term of the A.P equals the 3rd term of the G.P, find the relationship between d and r.
35. The 1st term of both an A.P and a G.P is 3. If the 2nd term of the A.P is equal to the 2nd term of the G.P, find the possible values of d and r.
36. If the sum of the first 3 terms of an A.P equals the product of the first 2 terms of a G.P (with same first term), find the relation between d and r.
37. If a is the first term common to both an A.P and a G.P, and the 5th term of the A.P equals the 3rd term of the G.P, determine d in terms of a and r.
38. The 2nd term of an A.P is equal to the geometric mean of the 1st and 2nd terms of a G.P. Find the relation between a, d and r.
39. The sum of first 3 terms of an A.P is equal to the sum of first 2 terms of a G.P with the same first term. Find the relation connecting d and r.
40. The 1st term of an A.P equals the 1st term of a G.P, and the 2nd term of the A.P equals the geometric mean of the 2nd and 3rd terms of the G.P. Express d in terms of r.

Answers

1. 102
2. 165
3. 1536
4. 1275
5. a = 2, d = 5
6. 2550
7. 13, 15, 17, 25
8. 56
9. 16
10. -19,682
11. 3
12. 400
13. 3
14. First term = -4
15. d = r² - 1
16. 55
17. 88
18. 20
19. 5, 8, 7
20. 12th term
21. 580
22. 0.333
23. 144
24. 1545
25. 20
26. 5 terms
27. Proven Formula
28. d = r - 1
29. 20
30. 21
31. d = a(r² − 1)
32. d = a(r² − 1)
33. d = a(r − 1)/2
34. d = a(r² − 1)/3
35. d = 3(r − 1); a = 3
36. 3a + 3d = a²r ⇒ d = (a²r − 3a)/3
37. d = (ar² − a)/4
38. a + d = √(a × ar) ⇒ d = a(√r − 1)
39. 3a + 3d = a + ar ⇒ 2a + 3d = ar ⇒ d = (ar − 2a)/3
40. a + d = √[(ar)(ar²)] = ar√r ⇒ d = a(r√r − 1)