30. Arithmetic Progression Problems Part 1

1. If nth term of a sequence is $$n^2 + 4,$$ find the sequence. Is this in A.P.?

2. Find the first five terms of the sequence for which $$t_1=3, t_2=5$$ and $$t_{n+2} = t_n + t_{n+1}.$$

3. How many terms are there in the A.P. $$2, 9, 16, 23, ..., 93$$?

4. How many terms are identical in the two A.P. $$2, 4, 6, ...$$ up to $$100$$ terms and $$3, 6, 9, ...$$ up to $$80$$ terms?

5. Write the sequence whose nth term is

1. $$3n + 1$$

2. $$\frac{n}{n + 1}$$

3. $$\frac{n^2}{n + 1}$$

6. If $$nth$$ term of a sequence is $$2an + b$$, where $$a$$ and $$b$$ are constants. Is this sequence is in A.P.?

7. Find the $$7th$$ term of the A.P.:math: -4, -1, 2, …

8. Find all positive integers of $$3$$ digits divisible by $$7$$.

9. Find the $$10th$$ term of the sequence whose $$7th$$ and $$12th$$ are $$34$$ and $$64$$ respectively.

10. Each of the sequences $$3, 5, 7, ...$$ and $$4, 9, 14, ...$$ are continued for $$40$$ terms. Find the identical terms.

11. Divide $$15$$ into three terms which are in A.P. and sum of their squares is $$83$$.

12. The sum of three numbers is $$12$$ and sum of their cubes is $$408$$ then find the numbers.

13. Divide $$20$$ into $$4$$ numbers which are in A.P., such that the product of first and fourth to the product of the second and third is $$2:3$$.

14. If the sum of $$n$$ terms of a series is $$5n^2 + 3n,$$ find the $$nth$$ term.

15. If the sum of $$n$$ terms of a series is $$an^2 + bn$$ where $$a$$ and $$b$$ are constants then show that the terms of the series form an A.P.

16. Find the sum of the series $$(a + b)^2 + (a^2 + b^2) + (a - b)^2 + ...$$ to $$n$$ terms.

17. The interior angles of a polygon are in A.P. The smallest angle is $$120^{\circ}$$ and the common difference is $$5^{\circ}.$$ Find the number of sides of the polygon.

18. Find the maximum sum of $$40, 38, 36, ...$$

19. Find the least value of n such that $$1 + 3 + 5 + ...$$ to $$n$$ terms $$\ge$$ 500.

20. If $$m$$ times the $$mth$$ term is equal to $$n$$ times the $$nth$$ term find the $$(m + n)th$$ term.

21. If $$a, b, c$$ be the $$pth, qth$$ and $$rth$$ terms respectively of an A.P., prove that $$a(q - r) + b(r - p) + c(p - q) = 0$$.

22. If $$a, b, c$$ are in A.P. prove that $$b + c, c + a, a + b$$ are also in A.P.

23. If $$a^2, b^2, c^2$$ are in A.P. prove that $$\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}$$ are in A.P.

24. If $$pth$$ term of an A.P. is $$q$$ and $$qth$$ term is $$p$$ then find the first term and common difference of that A.P. Also show that $$(p + q)th$$ term is zero.

25. For an A.P. show that $$t_m + t_{2n + m} = 2t_{m + n}.$$

26. If $$\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$$ are in A.P. and $$a + b + c \ne 0,$$ prove that $$\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}$$ are also in A.P.

27. If $$a, b, c$$ are in A.P. then prove that

1. $$\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}$$ are in A.P.

2. $$a^2(b + c), b^2(c + a), c^2(a + b)$$ are in A.P.

3. $$\frac{ab + ac}{bc}, \frac{bc + ba}{ca}, \frac{ca + bc}{ab}$$ are in A. P.

28. If $$\frac{b + c - a}{a}, \frac{c + a - b}{b}, \frac{a + b - c}{c}$$ are in A.P. the show that $$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$$ are in A.P. provided $$a + b + c \ne 0.$$

29. If $$(b - c)^2, (c - a)^2, (a - b)^2$$ are in A.P. then show that $$\frac{1}{b - c}, \frac{1}{c -a}, \frac{1}{a -b}$$ are in A.P.

30. If $$a^2(b + c), b^2(c + a), c^2(a + b)$$ are in A.P. then prove that $$a, b, c$$ are in A.P. or $$ab + bc + ca = 0.$$

31. If $$a$$ be the first term of an A.P. and the sum of its first $$p$$ terms is 0, show that the sum of next q terms is $$-\frac{aq(p + q)}{p - 1}.$$

32. If sum of the first $$p$$ terms of an A.P. is equal to sum of of its first $$q$$ terms, prove that the sum of its first $$(p + q)$$ terms is $$0$$.

33. Prove that the sum of the latter half of $$2n$$ terms of an A.P. is equal to one third of the sum of first $$3n$$ terms.

34. If $$S_1, S_2, S_3, ..., S_n$$ be the sums of $$n$$ arithmetic progressions whose first terms are $$1, 2, 3, ..., n$$ and common differences are $$1, 2, 3, ...$$ prove that

$S_1 + S_2 + ... + S_p = \frac{np}{4}(n + 1)(p + 1)$
35. If $$a, b, c$$ be the sum of $$p, q$$ and $$r$$ terms of an A.P. then prove that

$\frac{a}{p}(q - r) + \frac{b}{q}(r - p) + \frac{c}{r}(p - q) = 0.$
36. If the sum of $$m$$ terms of an A.P is equal to half the sum of $$(m + n)$$ terms and is also equal to half the sum of $$(m + p)$$ terms, prove that

$(m + n)\left(\frac{1}{m} - \frac{1}{p}\right) = (m + p)\left(\frac{1}{m} - \frac{1}{n}\right)$
37. If there are $$(2n + 1)$$ terms in an A.P. then prove that the ratio of sum odd terms and sum of even terms is $$(n + 1)/n$$.

38. The sum of $$n$$ terms of two series in A.P. are in the ratio $$(3n -13):(5n + 21)$$. Find the ratio of their $$24th$$ terms.

39. If the $$mth$$ term of an A.P. is $$\frac{1}{n}$$ and $$nth$$ term is $$\frac{1}{m}$$ the prove that the sum of $$mn$$ terms is $$\frac{mn + 1}{2}.$$

40. If the sum of $$m$$ terms of an A.P. is $$n$$ and the sum of its $$n$$ terms is $$m$$, show that the sum of $$(m + n)$$ terms is $$-(m + n)$$.

41. If the sum of $$n, 2n$$ and 3n terms of an A.P. be $$S_1, S_2$$ and $$S_3$$ respectively, show that $$S_3 = 3(S_2 - S_1).$$

42. If $$S$$ be the sum of $$(2n + 1)$$ terms of an A.P. and $$S_1$$ that of alternate terms beginning with their first, then show that

$\frac{S}{S_1} = \frac{2n + 1}{n + 1}$
43. If $$a, b$$ and $$c$$ be the $$1st, 3rd$$ and $$nth$$ terms respectively of an A.P., prove that the sum to $$n$$ terms is

$\frac{c + a}{2} + \frac{c^2 - a^2}{b - a}.$
44. If the $$pth$$ term of an A.P. is $$x$$ and $$qth$$ term is $$y$$. Show that the sum of $$(p + q)$$ terms is

$\frac{p + q}{2}\left\{x + y + \frac{x - y}{p - q}\right\}.$
45. The sum of $$n$$ terms of two series in A.P. are in the ratio $$(3n + 8):(7n + 15)$$. Find the ratio of their $$12th$$ term.

46. If the ratio of the sum of $$m$$ terms and $$n$$ terms of an A.P. is $$m^2:n^2,$$ prove that ratio of its $$mth$$ term and $$nth$$ term is $$(2m -1):(2n - 1)$$.

47. Show that any power $$n^k (k \ge 2),$$ and $$k$$ is an integer, can be represented in the form of a sum of $$n$$ successive odd numbers.

48. Let the sequence $$a_1, a_2, ..., a_n$$ form an arithmetic progression and $$a_1 = 0.$$ Simplify the expression

$S = \frac{a_3}{a_2} + \frac{a_4}{a_3} + ... + \frac{a_n}{a_{n - 1}} - a_2\left\{\frac{1}{a_2} + \frac{1}{a_3} + ... + \frac{1}{a_{n - 2}}\right\}.$
49. Prove that in any arithmetic progression $$a_1, a_2, a_3, ...$$ we have

$S = \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + ... + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_n}} = \frac{n - 1}{\sqrt{a_1} + \sqrt{a_n}}.$
50. Show that in any arithmetic progression $$a_1, a_2, a_3, ...$$ we have

$S = a_1^2 - a_2^2 + a_3^2 - a_4^2 + ... + a_{2k - 1}^2 - a_{2k}^2 = \frac{k}{2k - 1}(a_1^2 - a_{2k}^2).$