Geom series - Tn

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Sequences & Series - Geometric - the General term.
Test Yourself 1.

Finding a term	1. We are given the relationship that T_n = 3×2ⁿ. Find T₅. Answer.5th term is 96.	2. Find the 6^th term in the sequence 2, 10, 50, ... Answer.6th term is 6250.
	3. Find the 8^th term in the sequence -5, 20, -80, ... Answer.8th term is 81,920.	4. The first four terms of a geometric series are 3, x, y, 192. By finding the values of x and y, write out the first four terms in the series. Hint. You know the 1st and 4th terms! Answer.T₁ = 3 and r = 4. So 3, 12, 48, 192.
Finding a 1^st term	5. A geometric sequence has a third term of 18 and a common ratio of 3. Find the first term. Answer.First term is 2.	6. A sequence has a common ratio of 8 and a fifth term of 36,864. Find the first term. Answer.The 1st term is 9.
Finding a ratio	7. If the first term in a geometric series is 0.5 and the fifth term is 8, find the common ratio. Answer.ratio = 2.	8. If the first term in a geometric series is 1/3 and the sixth term is 2592, find the common ratio. Answer.ratio = 6.
Finding the number of terms	9. A geometric sequence has a first term of 32 and a common ratio of 1.5. Which term in this series equals 364.5? Answer.It is the 7th term.	10. A geometric sequence has a first term of - 0.5 and a common ratio of 2. Which term in this series equals -8? Answer.It is the 5th term.
Given 2 terms	11. The 3^rd term of a geometric series is 32 and the 7^th term is 8192. Find the first two terms of the series. Answer.T₁ = 2 and T₂ = 8.	12. The 4^th and 10^th terms of a geometric progression are 27 and 19,683 respectively. Find the first two terms of the sequence. Answer.T₁ = 1 and T₂ = 3.
	13. The third and seventh terms of a geometric series are 1.25 and 20 respectively. (i) Find the common ratio. (ii) Find the first term. (iii) Find the 13th term. Answer:(i) ratio = 2. (ii) T₁ = 5/16. (iii) T₁₃ = 1280.	14. If the 5^th term of a geometric series is 7 times the 4^th term and the second term is 7, what are the first three terms in the sequence? Answer:Sequence starts 1, 7, 49.
Miscellaneous	15. The sum of the 1^st and 2^nd terms of a geometric sequence is 72. The sum of the 3^rd and 4^th terms is 8. Find the first term and the common ratio for the two possible geometric sequences. Answer:x = 12 and y = 48.The series is 3, 12, 48 and 192.	16. The sum of the 3^rd and 4^th terms in a geometric sequence is 48. The difference between the 3^rd and 4^th terms in that geometric sequence is 24. Find the last term in this seqence which does not exceed 145. Answer:T₅ = 108.
	17. x + 1, x - 2 and 2x + 4 are in geometric progression. (i) Find the value(s) of x. (ii) Write out the first three terms of the sequence(s). Answer.First 2 terms are -9, -12, -16 OR 1, -2, 4.	18. 2x - 1, x + 1 and x - 1 are in geometric progression. Find the three terms of the sequence. Answer.First 2 terms are 9, 6, 4 OR -1, 1, -1.
	19. If a, b and 9 are in arithmetic progression and a, b and 12 are in geometric progression: (i) find possible values for a and b; (ii) determine the corresponding sequences using those values for a and b. Answer.a = 3 and b = 6 or a = 27 and b = 18.	20. The sequence is a geometric progression. For what values of x does the common ratio exceed 1? Answer.For x < 1 or for x > 1.