Chapter 9

If \(a, b, c\) are in A.P., prove that \(\frac{b c}{c a+a b}, \frac{c a}{b c+a b}, \frac{a b}{b c+c a}\) are in H.P.

First three of the four numbers are in A.P. \& the last three in H.P. Prove that the four numbers are proportional.

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

If \(x, y, z\) are in A.P., \(a x, b y, c z\) in G.P. and \(a, b, c\) in H.P. prove that \(\frac{x}{z}+\frac{z}{x}=\frac{a}{c}+\frac{c}{a}\).

If \(a^{x}=b^{y}=c^{z}\) and \(a, b, c\) be in G.P. then prove that \(x, y, z\) are in H.P.

If \(a, b, c\) be in G.P. then prove that \(\log _{a} n, \log _{b} n, \log _{c} n\) are in H.P.

Given \(a^{x}=b^{y}=c^{z}=d^{u}\) and \(a, b, c, d\) are in G.P., show that \(x, y, z, u\) are in H.P.

If \(a, b, c, d, e\) be five numbers such that \(a, b, c\) are in A.P., \(b, c, d\) are in G.P. and \(c, d, e\) are in H.P., prove that \(a, c, e\) are in G.P. and \(e=\frac{(2 b-a)^{2}}{a}\). If \(a=2\) and \(e=18\), find all possible values of \(b, c\) and \(d\).

\(a, b, c\) are in H.P., \(b, c, d\) are in G.P. and \(c, d, e\) are in A.P. show that \(e=\frac{a b^{2}}{(2 a-b)^{2}}\).

If \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\) and \(p, q, r\) be in A.P. then prove that \(x, y, z\) are in H.P.

If \(a, b, c\) are all positive and in H.P., then show that the roots of \(a x^{2}+2 b x+3 c\) are imaginary.

If \(a, b, c\) be in A.P. and \(a^{2}, b^{2}, c^{2}\) in H.P. then prove that either \(-\frac{a}{2}, b, c\) are in G.P. or \(a=b=c\).

\(p, q, r\) are three numbers in G.P. Prove that the first term of an A.P., whose \(p\) th, \(q\) th and \(r\) th terms are in H.P., is to the common difference as \((q+1): 1\).

A G.P. and a H.P. have the same \(p\) th, \(q\) th and \(r\) th terms as \(a, b, c\) respectively. Show that \(a(b-c) \log a+b(c-a) \log b+c(a-b) \log c=0 .\)

An A.P., a G.P and a H.P. have \(a\) and \(b\) for their first two terms. Show that their \((n+2)\) th terms will be in G.P. if \(\frac{b^{2 n+2}-a^{2 n+2}}{a b\left(b^{2 n}-a^{2 n}\right)}=\frac{n+1}{n}\).

An A.P. and a H.P., have the same first term, the same last term, and the same number of terms; prove that the product of the \(r\) th term from the beginning in one series and the \(r\) th term from the end in the other is independent of \(r\).

\(\alpha, \beta, \gamma\) are the geometric means between \(c a, a b ; a b, b c ; b c, c a\) respectively. Prove that if \(a, b, c\) are in A.P., then \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are also in A.P., and \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) are in H.P.

If the \((m+1)^{\text {th }},(n+1)^{\text {sh }}\) and \((r+1)^{t h}\) terms of an A.P. are in G.P., \(m, n, r\) are in H.P. show that the ratio of the common difference to the first term in the A.P. is \(-\frac{2}{n}\).

If \(S_{1}, S_{2}, S_{3}\) denote the sums of \(n\) terms of three A.P.'s whose first terms are unity and common differences in H.P., prove that \(n=\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\).

If \(a, b, c\) are in A.P., \(\alpha, \beta, \gamma\) in H.P., \(a \alpha, b \beta, c \gamma\) in G.P., (with common ratio not equal to \(1 .)\), then prove that \(a: b: c=\frac{1}{\gamma}: \frac{1}{\beta}: \frac{1}{\alpha}\).

Insert six harmonic means between 3 and \(\frac{6}{23}\).

Let the harmonic mean and geometric mean of two positive numbers be in the ratio \(4: 5\), then find the ratio of the two numbers.

If A.M between two numbers is 5 and their GM is 4, then find their HM.

A.M and H.M. between two quantities are 27 and 12 respectively, find their G.M.

If \(x, 1, z\) are in AP and \(x, 2, z\) are in GP, then find the harmonic mean of \(x\) and \(z\).

If \(H\) be the HM between \(a\) and \(b\), then show that \(\frac{H}{a}+\frac{H}{b}=2\).

The harmonic mean of two numbers is 4 . Their A.M., \(A\), and G.M., \(G\), satisfy the relation \(2 A+G^{2}=27\). Find the two numbers.

The A.M. of two numbers exceeds their G.M. by 15 and H.M. by 27 , find the numbers

If the A.M. between two numbers exceeds their G.M. by 2 and the G.M. exceeds their H.M. by \(\frac{8}{5}\); find the numbers.

If the A.M., the G.M and the H.M. of first and last terms of the sequence \(25,26,27, \ldots, N-1, N\) are the term of this sequence, find the value of \(N\).

If 9 arithmetic and harmonic means be inserted between 2 and 3, prove that \(A+\frac{6}{H}=5\) where \(A\) is any of the A.M.'s and \(H\) the corresponding H.M.

If \(H\) be the harmonic mean between \(a\) and \(b\) then prove that \(\frac{1}{H-a}+\frac{1}{H-b}=\frac{1}{a}+\frac{1}{b}\).

If \(A, G, H\) be respectively the A.M., G.M. and H.M. between two given quantities \(a\) and \(b\), then prove that \(A, G, H\) are in G.P.

If \(A\) be the A.M. and \(H\) the H.M. between two numbers \(a\) and \(b\) then \(\frac{a-A}{a-H} \times \frac{b-A}{b-H}=\frac{A}{H}\).

If \(A_{1}, A_{2} ; G_{1}, G_{2}\); and \(H_{1}, H_{2}\) be two A.M.'s and G.M.'s and H.M.'s between two quantities then prove that \(\frac{G_{1} G_{2}}{H_{1} H_{2}}=\frac{A_{1}+A_{2}}{H_{1}+H_{2}}\)

If \(n\) harmonic means are inserted between 1 and \(r\) then show that \(\frac{1 \text { st mean }}{n \text { th mean }}=\frac{n+r}{n r+1}\).

If \(H_{1}, H_{2}, \ldots H_{n}\) be \(n\) harmonic means between \(a\) and \(b\) show that \(\frac{H_{1}+a}{H_{1}-a}+\frac{H_{n}+b}{H_{n}-b}=2 n\). If \(n\) be a root of the equation \(x^{2}(1-a b)-x\left(a^{2}+b^{2}\right)-(1+a b)=0\), prove that \(H_{1}-H_{n}=a b(a-b)\).

For what value of \(n, \frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}\) is the harmonic mean of \(a\) and \(b ?\)

If \(a\) be A.M. of \(b\) and \(c, b\) the G.M. of \(c\) and \(a\), then prove that \(c\) is the \(\mathrm{H} . \mathrm{M}\). of \(a\) and \(b\).

If \(2(y-a)\) is the H.M. between \(y-x\) and \(y-z\), then show that \(x-a, y-a, z-a\) are in G.P.

If \(p\) be the first of \(n\) arithmetic means between two numbers and \(q\) be the first of \(n\) harmonic means

Sum the series \(1 \cdot 2^{2}+2 \cdot 3^{2}+3 \cdot 4^{2}+\cdots \cdots\) to \(n\) terms.

Sum the series \(1 \cdot 3^{2}+2 \cdot 5^{2}+3 \cdot 7^{2}+\cdots \cdots \cdot\) to 20 terms.

Sum the series \(1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5+\cdots \cdots \cdot\) to \(n\) terms.

Sum the series \(1 \cdot 2 \cdot 5+2 \cdot 3 \cdot 6+3 \cdot 4 \cdot 7+\cdots \cdots \cdot\) to \(n\) terms.

Sum the series \(1+(1+2)+(1+2+3)+(1+2+3+4)+\cdots \cdots \cdot\) to \(n\) terms.

Sum the series \(1^{2}+\left(1^{2}+2^{2}\right)+\left(1^{2}+2^{2}+3^{2}\right)+\cdots \cdots \cdot\) to \(n\) terms.

Sum the series \(\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\cdots \cdot .\) to 16 terms .

Find the sum of the series \(31^{3}+32^{3}+\cdots \cdots+50^{3}\).

Sum to \(n\) terms the series \(1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-6^{2}+\).