Chapter 9

If \(7^{\text {th }}\) and \(13^{\text {th }}\) terms of an AP be 34 and 64 respectively, then find its \(18^{\text {th }}\) term.

Divide 28 into four parts in A.P. so that ratio of the product of first and third with the product of second and fourth is \(8: 15\).

If \(x, a, b, c\) are real and \((x-a+b)^{2}+(x-b+c)^{2}=0\), then show that \(a, b, c\) are in AP.

Prove that if \(p, q, r(p \neq q)\) are terms (not necessarily consecutive) of an A.P., than there exists a rational number \(k\) such that \(\frac{(r-q)}{(q-p)}=k\).

Prove that the numbers \(\sqrt{2}, \sqrt{3}, \sqrt{5}\) cannot be the terms of a single A.P. with non-zero common difference.

Find the number of terms common to the two A.P.'s \(3,7,11, \ldots, 407\) and \(2,9,16, \ldots, 709\).

Prove that there are 17 identical terms in the two A.P.'s \(2,5,8,11, \ldots 60\) terms and \(3,5,7,9, \ldots 50\) terms.

If the roots of the equation \(x^{3}-12 x^{2}+39 x-28=0\) are in A.P., then find their common difference.

The \(m\) th term of an A. P. is \(n\) and its \(n\) th term is \(m\). Prove that its \(p\) th term is \(m+n+p\). Also show that its \((m+n)\) th term is zero.

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

If \(a, b, c\) are in A.P., then prove that \((a-c)^{2}=4\left(b^{2}-a c\right)\).

If \(a, b, c\) are in A.P., prove that the following are also in A.P.:- i. \(\frac{1}{b c}, \frac{1}{c a}, \frac{1}{a b}\). ii. \(b+c, c+a, a+b\). iii. \(a^{2}(b+c), b^{2}(c+a), c^{2}(a+b)\). iv. \(a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right), \quad c\left(\frac{1}{a}+\frac{1}{b}\right)\). v. \(\frac{1}{\sqrt{b}+\sqrt{c}}, \frac{1}{\sqrt{c}+\sqrt{a}}, \frac{1}{\sqrt{a}+\sqrt{b}}\).

If \(a^{2}, b^{2}, c^{2}\) are in A.P., then the following are also in A.P.:- i. \(\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b}\). ii. \(\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}\).

If \(\frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c}\) are in A.P., then \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are also in A.P.

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

If \(\log 2, \log \left(2^{x}-1\right)\) and \(\log \left(2^{x}+3\right)\) be three consecutive terms of an A.P., then find \(x\).

For what values of the parameter \(a\) are there values of \(x\) such that \(5^{1+x}+5^{1-x}, \frac{a}{2}, 25^{x}+25^{-x}\) are three consecutive terms of an A.P.?

If \(x^{18}=y^{21}=z^{28}\), prove that \(3,3 \log _{y} x, 3 \log _{z} y, 7 \log _{x} z\) form an A.P.

Each of the two triplets of numbers \(\log a, \log b, \log c\) and \(\log a-\log 2 b, \log 2 b-\log 3 c, \log 3 c-\log a\) are in A.P. Can the numbers \(a, b, c\) be the lengths of the sides of a triangle?

The fifth term of an A.P. is 1 whereas its 3 1st term is \(-77\). Find its 20 th term and sum of its first fifteen terms. Also find which term of the series will be \(-17\) and sum of how many terms will be 20 .

The third term of an A.P. is 7 and its 7 th term is 2 more than thrice of its 3 rd term. Find the first term, common difference and the sum of its first 20 terms.

Find the number of terms in the series \(20+19 \frac{1}{3}+18 \frac{2}{3}+\cdots \cdots\) of which the sum is 300 . Explain the double answer. Also find the maximum sum of the series.

How many terms of the sequence \(54,51,48, \ldots \ldots \ldots \ldots .\) be taken so that their sum is 513 ? Explain the double answer.

If the sum of three consecutive terms of an increasing AP is 51 and the product of the first and third of these terms is 273 , then find the third term.

Find the sum of all 2 digit odd numbers.

If the sum of the sequence \(2,5,8,11, \ldots \ldots \ldots .\) is 60100 , then find the number of terms.

The \(n\) th term of a series is given to be \(\frac{3+n}{4}\), find the sum of 105 terms of this series.

The first and last terms of an AP are 1 and 11 . If the sum of its terms is 36 , then find the number of terms.

If the sum of first 8 and 19 terms of an A.P. are 64 and 361 respectively, find the common difference and sum of its \(n\) terms.

A man arranges to pay off a debt of Rs. 3600 in 40 annual installments, which form an arithmetic series. When 30 of the installments are paid he dies leaving one-third of the debt unpaid. Find the value of the first installment.

A class consists of a number of boys whose ages are in A.P., the common difference being 4 months. If the youngest boy is just eight years old and if the sum of the ages is 168 years, find the number of boys.

The sum of \(n\) term of a series is \(3 n^{2}+4 n\). Show that the series is an A.P. and find the first term and common difference. What will be its \(n\) th term?

If the sum of \(n\) terms of an AP be \(3 n^{2}-n\), then find its first term and common difference.

If the sum of \(n\) terms of an AP is \(2 n^{2}+5 n\), then show that its \(n t h\) term is \(4 n+3\).

Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by \(3 .\)

If \(\frac{3+5+7+\ldots \ldots+n \text { terms }}{5+8+11+\ldots \ldots+10 \text { terms }}=7\), then find the value of \(n\).

Show that the sum of all odd numbers between 1 and 1000 , which are divisible by 3 , is \(83667 .\)

Find the sum of first \(n\) odd natural numbers.

Find the sum of all odd integers between 2 and 100 divisible by 3 .

Find the sum of all two digit numbers which when divided by 4 , yield unity as remainder.

Find the sum of integers from 1 to 100 which are divisible by 2 or 5 .

The sum of \(n\) terms of the two series \(3+10+17+\ldots \ldots\) and \(63+65+67+\ldots \ldots\) are equal, then find the value of \(n\).

The series of natural numbers is divided into groups \((1),(2,3,4),(3,4,5,6,7),(4,5,6,7,8,9,10), \ldots \ldots\) Find the sum of the numbers in \(n\) th group.

The series of natural numbers is dividend into groups (1); \((2,3,4) ;(5,6,7,8,9) ; \ldots \ldots\) and so on . Show that the sum of the numbers in the \(n\) th group is \((n-1)^{3}+n^{3}\).

\(N\), the set of natural numbers, is partitioned into subsets \(S_{1}=\\{1\\}, S_{2}=\\{2,3\\}, S_{3}=\\{4,5,6\\}, S_{4}=\\{7,8,9\) \(10\\}\). Find the sum of the elements in the subset \(S_{50}\).

The sum of three numbers in A.P. is 15 whereas sum of their squares is 83 . Find the numbers.

The sum of three numbers in A.P. is 12 and the sum of their cubes is 288 . Find the numbers.

Find four numbers in A.P. whose sum is 20 and sum of their squares is 120 .

Find four numbers in A.P. whose sum is 32 and sum of squares is 276 .

The number of terms of an A.P. is even; the sum of the odd terms is 24 , of the even terms 30 , and the last term exceeds the first by \(10 \frac{1}{2}\), find the number of terms and the series.