🧠 30-Second GP Recall
GP:
Each term is obtained by multiplying the previous term by the same non-zero number.
Each term is obtained by multiplying the previous term by the same non-zero number.
Common ratio:
r = second term ÷ first term
r = second term ÷ first term
n-th term from beginning:
an = arn−1
an = arn−1
n-th term from end:
If last term is l, Tn from end = lr−(n−1)
If last term is l, Tn from end = lr−(n−1)
Middle term:
For an odd number of terms, the middle term is T(N+1)/2.
For an odd number of terms, the middle term is T(N+1)/2.
Two middle terms:
For an even number of terms, positions are N/2 and N/2+1.
For an even number of terms, positions are N/2 and N/2+1.
Memory trick: “GP = × same number.”
To move forward, multiply by r. To move backward, divide by r.
To move forward, multiply by r. To move backward, divide by r.
A. Find the Common Ratio
1. Find r for: 3, 12, 48, 192, ...
2. Find r for: 81, 27, 9, 3, ...
3. Find r for: −2, 6, −18, 54, ...
4. Is 2, 6, 18, 54, 162 a GP? Give a reason.
5. Is 5, 10, 20, 41, 82 a GP? Explain without finding all terms.
B. n-th Term from the Beginning
6. Find the 8th term of 2, 6, 18, 54, ...
7. Find the 10th term of 5, 10, 20, 40, ...
8. Find the 7th term of 128, 64, 32, 16, ...
9. The first term of a GP is 3 and common ratio is 2. Find its 12th term.
10. The first term is 64 and common ratio is 1/2. Which term is 4?
C. n-th Term from the End
Recall: If the last term is l, move backward by dividing by r each time.
11. Find the 3rd term from the end of 2, 6, 18, 54, 162.
12. Find the 4th term from the end of 3, 9, 27, 81, 243, 729.
13. A GP has 8 terms. Its first term is 2 and common ratio is 3. Find the 5th term from the end.
14. A GP has 10 terms. Its last term is 1536 and common ratio is 2. Find the 6th term from the end.
D. Middle Term(s)
15. Find the middle term of: 2, 6, 18, 54, 162, 486, 1458.
16. Find the middle term of a GP having 9 terms, first term 3 and common ratio 2.
17. A GP has 11 terms. Which term is the middle term? Then write its position from the end.
18. A GP has 12 terms. Are there one or two middle terms? Name their positions from the beginning.
19. Find both middle terms of 2, 4, 8, 16, 32, 64, 128, 256.
E. 🔥 HOT Questions — Think, Don't Just Apply Formula
20. Three consecutive terms of a GP are x, 12, 48. Find x.
21. Three consecutive terms of a GP are x−2, x, x+8. Find x.
22. The 3rd term of a GP is 12 and the 6th term is 96. Find the common ratio, assuming it is positive.
23. The 4th term of a GP is 24 and the 7th term is 192. Find the first term and common ratio.
24. A GP has first term 5 and common ratio 3. One of its terms is 405. Which term is it?
25. In a GP, the 5th term is 81 and the 8th term is 2187. Find r and the first term.
26. A GP has 9 terms. The first term is 2 and the last term is 512. Find the middle term without writing all the terms.
27. The 4th term from the beginning is equal to the 4th term from the end of a GP with 7 terms. What does this tell you about the sequence?
F. MCQ Revision
28. Which is a GP?
(A) 2, 4, 8, 16(B) 2, 4, 7, 11(C) 3, 6, 10, 15(D) 1, 3, 6, 10
29. The common ratio of 81, 27, 9, 3 is:
(A) 3(B) 1/3(C) −3(D) −1/3
30. The 6th term of 2, 4, 8, ... is:
(A) 16(B) 32(C) 64(D) 128
31. A GP has 15 terms. Its middle term is:
(A) 7th(B) 8th(C) 7th and 8th(D) 15th
32. The 3rd term from the end of 3, 6, 12, 24, 48 is:
(A) 6(B) 12(C) 24(D) 48
33. If a=5, r=−2, then the 4th term is:
(A) −40(B) 40(C) −20(D) 20
34. If the first term is 4 and the common ratio is 3, which term is 108?
(A) 3rd(B) 4th(C) 5th(D) 6th
35. If three consecutive terms of a GP are x/r, x, xr, their product is:
(A) x(B) x²(C) x³(D) r³
36. For a GP with 9 terms, the 4th term from the beginning and 6th term from the end are:
(A) same term(B) adjacent terms(C) first and last(D) cannot be related
G. Final Recall Challenge
Q1. What is the fastest way to test whether a sequence is GP?
Q2. How do you move one position toward the beginning?
Q3. For 13 terms, which term is in the middle?
Q4. For 14 terms, how many middle terms are there?
🔒 Teacher Answer Key — hidden from students
1. 4
2. 1/3
3. −3
4. Yes, ratio = 3.
5. No; 41/20 ≠ 2.
6. 4374
7. 2560
8. 2
9. 6144
10. 5th term
11. 54
12. 27
13. 54
14. 48
15. 54
16. 48
17. 6th term; also 6th from end.
18. Two middle terms: 6th and 7th.
19. 16 and 32.
20. 3
21. x = 4
22. r = 2; first term = 3.
23. r = 2; first term = 3.
24. 5th term.
25. r = 3; first term = 1.
26. 32.
27. The 4th term is the middle term; the sequence has 7 terms and is symmetric in position about the middle.
28. A 29. B 30. B 31. B 32. B 33. A 34. B 35. C 36. A
Final Recall: Q1: Divide consecutive terms and check the ratio is constant. Q2: Divide by r. Q3: 7th. Q4: 2.
Teacher use: keep the answer key collapsed when projecting the sheet. Students can complete the questions without seeing the solutions.