# 23. Arithmetic Progression Problems Part 1¶

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

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

How many terms are there in the A.P. \(2, 9, 16, 23, ..., 93\)?

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?

Write the sequence whose nth term is

- \(3n + 1\)
- \(\frac{n}{n + 1}\)
- \(\frac{n^2}{n + 1}\)

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

Find the \(7th\) term of the A.P.:math:` -4, -1, 2, …`

Find all positive integers of \(3\) digits divisible by \(7\).

Find the \(10th\) term of the sequence whose \(7th\) and \(12th\) are \(34\) and \(64\) respectively.

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

Divide \(15\) into three terms which are in A.P. and sum of their squares is \(83\).

The sum of three numbers is \(12\) and sum of their cubes is \(408\) then find the numbers.

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 \(2:3\).

If the sum of \(n\) terms of a series is \(5n^2 + 3n,\) find the \(nth\) term.

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.

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

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.

Find the maximum sum of \(40, 38, 36, ...\)

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

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

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

If \(a, b, c\) are in A.P. prove that \(b + c, c + a, a + b\) are also in A.P.

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.

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.

For an A.P. show that \(t_m + t_{2n + m} = 2t_{m + n}.\)

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.

If \(a, b, c\) are in A.P. then prove that

- \(\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}\) are in A.P.
- \(a^2(b + c), b^2(c + a), c^2(a + b)\) are in A.P.
- \(\frac{ab + ac}{bc}, \frac{bc + ba}{ca}, \frac{ca + bc}{ab}\) are in A. P.

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 \(a + b + c \ne 0.\)

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.

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 \(ab + bc + ca = 0.\)

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 \(-\frac{aq(p + q)}{p - 1}.\)

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 \(0\).

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.

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

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

\[\frac{a}{p}(q - r) + \frac{b}{q}(r - p) + \frac{c}{r}(p - q) = 0.\]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

\[(m + n)\left(\frac{1}{m} - \frac{1}{p}\right) = (m + p)\left(\frac{1}{m} - \frac{1}{n}\right)\]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 \((n + 1)/n\).

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.

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 \(\frac{mn + 1}{2}.\)

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 \(-(m + n)\).

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

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

\[\frac{S}{S_1} = \frac{2n + 1}{n + 1}\]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

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

\[\frac{p + q}{2}\left\{x + y + \frac{x - y}{p - q}\right\}.\]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.

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 \((2m -1):(2n - 1)\).

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.

Let the sequence \(a_1, a_2, ..., a_n\) form an arithmetic progression and \(a_1 = 0.\) Simplify the expression

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

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

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