28. Geometric Progressions Problems Part 1¶

1. How many terms are there in the G. P. $$5, 20, 80, ..., 5120$$?

2. If the fifth term of a G. P. is 81 and second term is 24, find the G. P.

3. How many terms are there in the G. P. $$0.03, 0.06, 0.12, ..., 3.84$$.

4. If the population of a city on $$1st$$ January $$2006$$ was $$20000$$. If it increased at the rate of $$2\%$$ per annum, find the population of the city ion $$1st$$ January $$2016$$.

5. A person agrees to work at the rate of one dollar the first day, two dollars the second day, four dollars the third day, eight dollars the fourth day and so on in any month. How much would he get on $$20th$$ of that month?

6. The sum of $$n$$ terms of a sequence is $$2^n - 1$$, find its $$n$$ th term. Is this sequence a G. P.?

7. The seventh term of a G. P. is $$8$$ times the fourth term. Find the G. P. when its fifth term is 48.

8. If the product of three numbers in G. P. is $$216$$ and their sum is $$19$$, find th numbers.

9. A number consists of three digits in G. P. The sum of the left hand and right hand digits exceed twice the middle digit by $$1$$ and the sum of the left hand and middle digits is two-third the sum of the middle and right hand digits. Find the numbers.

10. In a set of four numbers, the first three are in G. P. and the last three are in A. P. with a common difference of 6. If the first number is same as the fourth, find the four numbers.

11. The sum of three numbers in G. P. is 21 and the sum of their squares is 189, find the numbers.

12. If sixth and tenth term of a G. P. are $$\frac{1}{16}$$ and $$\frac{1}{256}$$ respectively, then find the G. P.

13. If fifth and eighth term of a G. P. are $$48$$ and $$384$$ respectively, then find the G. P.

14. The product of three consecutive terms of a G. P. is $$-64$$ and the first term is four times the third. Find the G. P.

15. Find $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}+ ...$$ to $$n$$ terms.

16. Find the sum of $$n$$ terms of the series

$(a + b) + (a^2 + 2b) + (a^3 + 3b) + ...$
17. Find the sum to $$n$$ terms of the series $$7 + 77 + 777 + 7777 + ...$$

18. If you had a choice of a salary of $$\1000$$ a day for a month for $$31$$ days or $$\1$$ for the first day, $$\2$$ for the second day, $$\4$$ for the third day and each day thereafter for the rest of the month your salary would be doubled, which choice would you make?

19. How many terms of the series $$1 + 3 + 3^2 + 3^3 + ...$$ must be taken to make 3280?

20. Find the least value of $$n$$ for which $$1 + 3 + 3^2 + ... + 3^{n - 1} > 1000$$

21. Express $$0.4\dot{2}\dot{3}$$ as a rational number.

22. After striking the floor a rubber ball rebounds to $$\frac{4}{5}$$ th of the height from which it has fallen. Find the total distance it travels before coming to rest if it was dropped from a height of 120 meters.

23. Find the sum of the series $$1 - 3 + 9 - 27 + ...$$ to $$9$$ terms.

24. Evaluate $$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + ...$$ to $$n$$ terms.

25. Evaluate $$6 + 66 + 666 + ...$$ to $$n$$ terms.

26. Find the sum of $$n$$ terms for the series $$4 + 44 + 444 + ...$$

27. Find the value of $$0.5 + 0.55 + 0.555 + 0.5555 + ...$$ to $$n$$ terms.

28. Find the sum of the following series till $$n$$ terms:

$\left(x^2 + \frac{1}{x^2} + 2\right) + \left(x^3 + \frac{1}{x^3} + 5\right) + \left(x^3 + \frac{1}{x^3} + 8 + ...\right)$
29. If the $$(p + q)\text{th}$$ term of a G. P. is $$a$$ and $$(p - q)\text{th}$$ term is $$b$$, show that $$p\text{th}$$ term is $$\sqrt{ab}$$.

30. If the $$p\text{th}, q\text{th}$$ and $$r\text{th}$$ terms of a G. P. are $$x, y$$ and $$z$$ respectively, prove that $$x^{q - r}y^{r - p}z^{p - q} = 1$$.

31. Three numbers whose sum is 15 are in A. P. If $$1, 4, 19$$ be added to them the respectively then the resulting numbers are in G. P. Find the numbers.

32. From three numbers in G. P. other three numbers in G. P. are subtracted and the result is also found to be in G. P. then prove that the three sequences have the same common ratio.

33. If the $$p\text{th}, q\text{th}$$ and $$r\text{th}$$ terms of a G. P. are $$a, b$$ and $$c~(a, b, c > 0)$$, then prove that

$(q - r)\log a + (r - p)\log b + (p - q)\log c = 0$
34. If $$a, b, c, d$$ are in G. P., then show that

$(b - c)^2 + (c - a)^2 + (d - b)^2 = (a - d)^2$
35. If $$a, b, c, d$$ are in G. P., then show that

$(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2$
36. If $$a^x = b^y = c^z$$ and $$x, y, z$$ are in G. P., show that $$\log_b a = \log_c b$$.

37. If $$a, b, c, d$$ are in G. P., show that $$(a + b)^2, (b + c)^2, (c + d)^2$$ are in G. P.

38. If $$a, b, c, d$$ are in G. P., show that $$(a - b)^2, (b - c)^2, (c - d)^2$$ are in G. P.

39. If $$a, b, c, d$$ are in G. P., show that $$a^2 + b^2 + c^2, ab + bc + cd, b^2 + c^2 + d^2$$ are in G. P.

40. If $$a, b, c, d$$ are in G. P., show that $$\frac{1}{(a + b)^2}, \frac{1}{(b + c)^2}, \frac{1}{(c + d)^2}$$ are in G. P.

41. If $$a, b, c, d$$ are in G. P., show that $$a(b - c)^3 = d(a - b)^3$$

42. If $$a, b, c, d$$ are in G. P., show that $$(a + b + c + d)^2 = (a + b)^2 + (c + d)^2 + 2(b + c)^2$$

43. If $$a, b, c$$ are in G. P., show that $$a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = a^3 + b^3 + c^3$$

44. If $$a, b, c, d$$ are in G. P., show that $$(a^2 - b^2)(b^2 + c^2) = (b^2 - c^2)(a^2 + b^2)$$

45. If $$S$$ be the sum, $$P$$ the product and $$R$$ the sum of reciprocals of $$n$$ terms in G. P., prove that

$P^2 = \left(\frac{S}{R}\right)^n$
46. Find $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} ...$$ to $$\infty$$.

47. Prove that in an infinite G. P. whose common ratio $$r$$ has the property, $$|r| < 1$$, the ratio of any terms to the sum of all succeeding terms is

$\frac{1 - r}{r}$
48. If $$S_1, S_2, S_3, ..., S_p$$ are the sums of infinite geometric series whose first terms are $$1, 2, 3, ..., p$$ and whose common ratios are $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ..., \frac{1}{p + 1}$$ respectively, prove that

$S_1 + S_2 + ... + S_p = \frac{p(p + 3)}{2}.$
49. If $$x = 1 + a + a + a^2 + a^3 + ...$$ to $$\infty$$ and $$y = 1 + b + b^2 + b^3 + ...$$ to $$\infty$$. Show that $$1 + ab + a^2b^2 + a^3b^3 + ...$$ to $$\infty = \frac{xy}{x + y - 1}$$ where, $$0 < a< 1$$ and $$0< b < 1$$.

50. Find the sum to infinity of the series $$1 + (1 + a)r + (1 + a + a^2)r^2 + ...$$ where, $$0 < a < 1$$ and $$0 < r < 1$$.