Arithmetic Progression Problems Part 1 ************************************** 1. If nth term of a sequence is :math:`n^2 + 4,` find the sequence. Is this in A.P.? 2. Find the first five terms of the sequence for which :math:`t_1=3, t_2=5` and :math:`t_{n+2} = t_n + t_{n+1}.` 3. How many terms are there in the A.P. :math:`2, 9, 16, 23, ..., 93`? 4. How many terms are identical in the two A.P. :math:`2, 4, 6, ...` up to :math:`100` terms and :math:`3, 6, 9, ...` up to :math:`80` terms? 5. Write the sequence whose nth term is (i) :math:`3n + 1` (ii) :math:`\frac{n}{n + 1}` (iii) :math:`\frac{n^2}{n + 1}` 6. If :math:`nth` term of a sequence is :math:`2an + b`, where :math:`a` and :math:`b` are constants. Is this sequence is in A.P.? 7. Find the :math:`7th` term of the A.P.:math:` -4, -1, 2, ...` 8. Find all positive integers of :math:`3` digits divisible by :math:`7`. 9. Find the :math:`10th` term of the sequence whose :math:`7th` and :math:`12th` are :math:`34` and :math:`64` respectively. 10. Each of the sequences :math:`3, 5, 7, ...` and :math:`4, 9, 14, ...` are continued for :math:`40` terms. Find the identical terms. 11. Divide :math:`15` into three terms which are in A.P. and sum of their squares is :math:`83`. 12. The sum of three numbers is :math:`12` and sum of their cubes is :math:`408` then find the numbers. 13. Divide :math:`20` into :math:`4` numbers which are in A.P., such that the product of first and fourth to the product of the second and third is :math:`2:3`. 14. If the sum of :math:`n` terms of a series is :math:`5n^2 + 3n,` find the :math:`nth` term. 15. If the sum of :math:`n` terms of a series is :math:`an^2 + bn` where :math:`a` and :math:`b` are constants then show that the terms of the series form an A.P. 16. Find the sum of the series :math:`(a + b)^2 + (a^2 + b^2) + (a - b)^2 + ...` to :math:`n` terms. 17. The interior angles of a polygon are in A.P. The smallest angle is :math:`120^{\circ}` and the common difference is :math:`5^{\circ}.` Find the number of sides of the polygon. 18. Find the maximum sum of :math:`40, 38, 36, ...` 19. Find the least value of n such that :math:`1 + 3 + 5 + ...` to :math:`n` terms :math:`\ge` 500. 20. If :math:`m` times the :math:`mth` term is equal to :math:`n` times the :math:`nth` term find the :math:`(m + n)th` term. 21. If :math:`a, b, c` be the :math:`pth, qth` and :math:`rth` terms respectively of an A.P., prove that :math:`a(q - r) + b(r - p) + c(p - q) = 0`. 22. If :math:`a, b, c` are in A.P. prove that :math:`b + c, c + a, a + b` are also in A.P. 23. If :math:`a^2, b^2, c^2` are in A.P. prove that :math:`\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}` are in A.P. 24. If :math:`pth` term of an A.P. is :math:`q` and :math:`qth` term is :math:`p` then find the first term and common difference of that A.P. Also show that :math:`(p + q)th` term is zero. 25. For an A.P. show that :math:`t_m + t_{2n + m} = 2t_{m + n}.` 26. If :math:`\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}` are in A.P. and :math:`a + b + c \ne 0,` prove that :math:`\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}` are also in A.P. 27. If :math:`a, b, c` are in A.P. then prove that (i) :math:`\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}` are in A.P. (ii) :math:`a^2(b + c), b^2(c + a), c^2(a + b)` are in A.P. (iii) :math:`\frac{ab + ac}{bc}, \frac{bc + ba}{ca}, \frac{ca + bc}{ab}` are in A. P. 28. If :math:`\frac{b + c - a}{a}, \frac{c + a - b}{b}, \frac{a + b - c}{c}` are in A.P. the show that :math:`\frac{1}{a}, \frac{1}{b}, \frac{1}{c}` are in A.P. provided :math:`a + b + c \ne 0.` 29. If :math:`(b - c)^2, (c - a)^2, (a - b)^2` are in A.P. then show that :math:`\frac{1}{b - c}, \frac{1}{c -a}, \frac{1}{a -b}` are in A.P. 30. If :math:`a^2(b + c), b^2(c + a), c^2(a + b)` are in A.P. then prove that :math:`a, b, c` are in A.P. or :math:`ab + bc + ca = 0.` 31. If :math:`a` be the first term of an A.P. and the sum of its first :math:`p` terms is 0, show that the sum of next q terms is :math:`-\frac{aq(p + q)}{p - 1}.` 32. If sum of the first :math:`p` terms of an A.P. is equal to sum of of its first :math:`q` terms, prove that the sum of its first :math:`(p + q)` terms is :math:`0`. 33. Prove that the sum of the latter half of :math:`2n` terms of an A.P. is equal to one third of the sum of first :math:`3n` terms. 34. If :math:`S_1, S_2, S_3, ..., S_n` be the sums of :math:`n` arithmetic progressions whose first terms are :math:`1, 2, 3, ..., n` and common differences are :math:`1, 2, 3, ...` prove that .. math:: S_1 + S_2 + ... + S_p = \frac{np}{4}(n + 1)(p + 1) 35. If :math:`a, b, c` be the sum of :math:`p, q` and :math:`r` terms of an A.P. then prove that .. math:: \frac{a}{p}(q - r) + \frac{b}{q}(r - p) + \frac{c}{r}(p - q) = 0. 36. If the sum of :math:`m` terms of an A.P is equal to half the sum of :math:`(m + n)` terms and is also equal to half the sum of :math:`(m + p)` terms, prove that .. math:: (m + n)\left(\frac{1}{m} - \frac{1}{p}\right) = (m + p)\left(\frac{1}{m} - \frac{1}{n}\right) 37. If there are :math:`(2n + 1)` terms in an A.P. then prove that the ratio of sum odd terms and sum of even terms is :math:`(n + 1)/n`. 38. The sum of :math:`n` terms of two series in A.P. are in the ratio :math:`(3n -13):(5n + 21)`. Find the ratio of their :math:`24th` terms. 39. If the :math:`mth` term of an A.P. is :math:`\frac{1}{n}` and :math:`nth` term is :math:`\frac{1}{m}` the prove that the sum of :math:`mn` terms is :math:`\frac{mn + 1}{2}.` 40. If the sum of :math:`m` terms of an A.P. is :math:`n` and the sum of its :math:`n` terms is :math:`m`, show that the sum of :math:`(m + n)` terms is :math:`-(m + n)`. 41. If the sum of :math:`n, 2n` and 3n terms of an A.P. be :math:`S_1, S_2` and :math:`S_3` respectively, show that :math:`S_3 = 3(S_2 - S_1).` 42. If :math:`S` be the sum of :math:`(2n + 1)` terms of an A.P. and :math:`S_1` that of alternate terms beginning with their first, then show that .. math:: \frac{S}{S_1} = \frac{2n + 1}{n + 1} 43. If :math:`a, b` and :math:`c` be the :math:`1st, 3rd` and :math:`nth` terms respectively of an A.P., prove that the sum to :math:`n` terms is .. math:: \frac{c + a}{2} + \frac{c^2 - a^2}{b - a}. 44. If the :math:`pth` term of an A.P. is :math:`x` and :math:`qth` term is :math:`y`. Show that the sum of :math:`(p + q)` terms is .. math:: \frac{p + q}{2}\left\{x + y + \frac{x - y}{p - q}\right\}. 45. The sum of :math:`n` terms of two series in A.P. are in the ratio :math:`(3n + 8):(7n + 15)`. Find the ratio of their :math:`12th` term. 46. If the ratio of the sum of :math:`m` terms and :math:`n` terms of an A.P. is :math:`m^2:n^2,` prove that ratio of its :math:`mth` term and :math:`nth` term is :math:`(2m -1):(2n - 1)`. 47. Show that any power :math:`n^k (k \ge 2),` and :math:`k` is an integer, can be represented in the form of a sum of :math:`n` successive odd numbers. 48. Let the sequence :math:`a_1, a_2, ..., a_n` form an arithmetic progression and :math:`a_1 = 0.` Simplify the expression .. math:: 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 :math:`a_1, a_2, a_3, ...` we have .. math:: 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 :math:`a_1, a_2, a_3, ...` we have .. math:: 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).