Geom series - limiting sum TYS 1

The questions on this page focus on:

1. using the limiting sum relationship;

2. conditions for a limiting sum.

Using the limiting sum relationship.	1. Find the limiting sum of the series: 24 + 12 + 6 + ... AnswerLimiting sum = 48.	2. Find the limiting sum of the series: AnswerLimiting sum = 27/8.
Find r.	3. An infinite geometric series has a limiting sum of 24.The first term is 15. Find the common ratio. Answer3/8.	4. An infinite geometric series has a first term of 8 and a limiting sum of 12. Find the common ratio. Answer:Ratio is 1/3.
	5. If the second term in a geometric series is 5 and the limiting sum is 20, find the first term and the common ratio. AnswerT₁=10 and r = 0.5.	6. Find the value(s) of x for which the terms 2 + x + 3 form a geometric series. Answer:Ratio = +√6 or - √6.
Find T₁.	7. In a particular geometric series, the common ratio is 0.25 and the limiting sum is 16. Determine the first term. Answer:T₁ = 12.	8. A geometric series has a first term of a and a limiting sum of 2. Find all possible values for a. Answer: 0 < a < 4.
Conditions for a limiting sum.	9. For what values of x will the series (1 + x) + (1 + x)² + (1 + x)³ + ... have a limiting sum? Answer-2 < x < 0.	10. (i) Show that the infinite series is a geometric series. (ii) Find the values such that this series has a limiting sum. (iii) Find the limiting sum in terms of x. Answer(i) r = 2x/3. (ii) -1.5 < x < 1.5. (iii) Limiting sum = x/(3 - 2x).
	11. Find the values for x for which the series 1 + sin 2x + sin² 2x + ... has a limiting sum. Answer-π/4 < x < π/4.	12. For what values of x will the following series have a limiting sum: log₁₀ x + (log₁₀ x)² + (log₁₀ x)³ + ... Answere^-1 < x < e.
	13. Consider the geometric series (i) Explain why the geometric series has a limiting sum. (ii) Find the exact value of the limiting sum. Write your answer with a rational denominator. Answer(i) √5 - 2 = 0.24 (approx) < 1. (ii) Limiting sum = (3 + √5)/4 .	14. 1 + 2x + 4x² + ... is a geometric series. (i) for what values of x does a limiting sum exist? (ii) Is it possible for the limiting sum to be 12? Give reasons. Answer:(i) -1/2 < x < 1/2 but x &ne. 0 Yes a limiting sum of 12 is possible if x = 11/24.
	15. (i) For what values of b will the infinite geometric series 1 + b + b² + ... have a limiting sum? (ii) Calculate the value of b if the limiting sum of 1 + b + b² ... is 0.8. Answer(i) -1 < b < 1. (ii) b = -0.25.	16. A series is 1 + 3tan²2x + 9tan⁴ 2x + 27 tan⁶2x + ... (i) For what values of x does the series have a limiting sum? (ii) Find the limiting sum of the series when x is in the middle of the positive half of the allowable range in (i). Answer:(i) -π/12 < x < π/12 but x &ne. 0 as all terms after the first = 0 (ii) When x = π/24, the limiting sum is 1.275 (approx).
Recurring decimals.	17. By considering the recurring decimal as the sum of an infinite geometric series, express in the form of a fraction. Answer1.	18. By considering the recurring decimal as the sum of an infinite geometric series, express in the form of a fraction. Answer41/90.
	19. By considering the recurring decimal as the sum of an infinite geometric series, express in the form of a fraction. Answer23/99.	20. Write the recurring decimal as a fraction by considering it as the sum of an infinite series. Answer.41/333.
Mixture	21. Find a number which, when added to each of the numbers 2, 6 and 13, results in three numbers in a geometric sequence. Answer.Number is 10/3.	22. Given that the limiting sums S₁ and S₂ of the series both exist where S₁ = 1 + sin² x + sin⁴ x + sin⁶ x +... S₂ = 1 + cos² x + cos⁴ x + cos⁶ x + ... (i) Show that S₁ = sec² x and S₂ = cosec² x. (ii) Show that S₁ + S₂ = S₁S₂.
Applied questions.	23. Jim is a keen gardener. Three years ago, he planted an Australian flowering gum when it was 120 cm tall. At the end of the first year after planting, it was 228 cm tall and it continued to grow at this rate of 90% of the previous growth per year. (i) By how much did the flowering gum grow in the third year? (ii) How tall (in metres) was the flowering gum after 5 years (to the nearest cm)? (iii) Assuming the gum tree maintains its growth rate, will it ever reach its advertised height of 12 m? Answer(i) Grew by 87.48 cm in the 3rd year. (ii) Height after 5 years is 5.62 m. (iii) 1200 cm (i.e. max height is 12 m) so yes!
	24. A weight lifter in training tires with each lift such that only 90% of the preceding weight can be lifted. (i) What will the weight lifter lift on the 5th attempt? (ii) Theoretically, what would be the total of the weights lifted by the weight lifter before becoming totally exhausted? Interpret your answer. Answer.10 times the first weight.	25. In the infinite geometric series 1 + x + x² + x³ + ... the first term is four times the sum of all terms following it. Find the value of x. Answerx = 0.2