Chapter 9

Show that the sum of \(n\) terms of a G.P. of common ratio \(r\) beginning with the \(p^{t h}\) term is \(r^{p-q}\) times the sum of an equal number of terms of the same series beginning with \(q^{t h}\) term.

Find the sum of \(2 n\) term of a series of which every even term is \(a\) times the term before it, and every odd term \(c\) times the term before it, the first term being unity.

If \(S\) is the sum to infinity of a GP, whose first term is \(a\), then find the sum of the first \(n\) terms.

If \(S\) be the sum, \(P\) the product and \(R\) the sum of the reciprocals of \(n\) terms of a G.P., prove that \(\left(\frac{S}{R}\right)^{n}=P^{2}\).

If \(S=1+a+a^{2}+\ldots \ldots\) to \(\infty(a<1)\), then show that \(a=\frac{S-1}{S}\).

If the sum of the series \(1+\frac{3}{x}+\frac{9}{x^{2}}+\frac{27}{x^{3}}+\ldots\) to \(\infty\) is a finite number, then show that \(x>3\).

Prove that \(\sum_{r=1}^{n} \log \left(\frac{a^{r}}{b^{r-1}}\right)=\frac{n}{2} \log \left(\frac{a^{n+1}}{b^{n-1}}\right)\)

If \(A=1+r^{a}+r^{2 a}+\ldots \ldots \ldots\) to \(\infty\) and \(B=1+r^{b}+r^{2 b}+\ldots \ldots \ldots\) to \(\infty\), prove that \(r=\left(\frac{A-1}{A}\right)^{1 / a}=\left(\frac{B-1}{B}\right)^{1 / b}\)

If \(S_{1}, S_{2}, S_{3}\) be respectively the sums of \(n, 2 n, 3 n\) terms of a G.P., then prove that i. \(\quad S_{1}\left(S_{3}-S_{2}\right)=\left(S_{2}-S_{1}\right)^{2}\) ii. \(\quad S_{1}^{2}+S_{2}^{2}=S_{1}\left(S_{2}+S_{3}\right)\).

If \(S_{1}, S_{2}, \ldots, S_{n}\) are the sums of infinite geometric series whose first terms are \(1,2,3, \ldots n\) and common ratios are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{n+1}\) respectively then prove that \(S_{1}+S_{2}+S_{3}+\ldots+S_{n}=\frac{1}{2} n(n+3)\).

If \(S_{p}\) denotes the sum of series \(1+r^{p}+r^{2 p}+\ldots \ldots\) to \(\infty\) and \(s_{p}\) the sum of the series \(1-r^{p}+r^{2 p}-\ldots \ldots \ldots\) to \(\infty\), prove that \(S_{p}+s_{p}=2 S_{2 p}\)

If \(S_{n}\) represents the sum of \(n\) terms of a G.P. whose first term and common ratio are \(a\) and \(r\) respectively, then prove that i. \(\quad S_{1}+S_{2}+S_{3}+\ldots+S_{n}=\frac{n a}{1-r}-\frac{\operatorname{ar}\left(1-r^{n}\right)}{(1-r)^{2}}\); ii. \(S_{1}+S_{3}+S_{5}+\ldots+S_{2 n-1}=\frac{a n}{1-r}-\frac{a r\left(1-r^{2 n}\right)}{(1-r)^{2}(1+r)}\).

The sum of the squares of three distinct real numbers, which are in G.P. is \(S^{2}\). If their sum is \(\alpha S\), show that \(\alpha^{2} \in\left(\frac{1}{3}, 1\right) \cup(1,3)\).

Find the natural number \(a\) for which \(\sum_{k=1}^{n} f(a+k)=16\left(2^{n}-1\right)\), where the function \(f\) satisfies the relation \(f(x+y)=f(x) \cdot f(y)\) for all natural numbers \(x, y\) and further \(f(1)=2 .\)

Sum the series \((a+b)+\left(a^{2}+2 b\right)+\left(a^{3}+3 b\right)+\ldots\) to \(n\) terms.

Sum the series \(\left(x+\frac{1}{x}\right)^{2}+\left(x^{2}+\frac{1}{x^{2}}\right)^{2}+\left(x^{3}+\frac{1}{x^{3}}\right)^{2} \ldots\left(x^{n}+\frac{1}{x^{n}}\right)^{2}\).

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

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

If \(x=1+a+a^{2}+a^{3}+\ldots\) to \(\infty(|a|<1)\) and \(y=1+b+b^{2}+b^{3}+\ldots \ldots\) to \(\infty(|b|<1)\). Prove that \(1+a b+a^{2} b^{2}+a^{3} b^{3}+\ldots\) to \(\infty=\frac{x y}{x+y-1} .\)

Sum the series \(1+\frac{\sqrt{2}-1}{2 \sqrt{3}}+\frac{3-2 \sqrt{2}}{12}+\frac{5 \sqrt{2}-7}{24 \sqrt{3}}+\frac{17-12 \sqrt{2}}{144}+\ldots \infty\)

Let \(r\) be the common ratio of the G.P. \(a_{1}, a_{2}, \ldots\), show that \(\frac{1}{a_{1}^{\mathrm{m}}+a_{2}^{m}}+\frac{1}{a_{2}^{m}+a_{3}^{m}}+\ldots+\frac{1}{a_{n-1}^{\mathrm{m}}+a_{n}^{\mathrm{m}}}=\frac{1-r^{m(1-n)}}{a_{1}^{m}\left(r^{m}-r^{-m}\right)}\)

Find the sum of the infinite series \(1+(1+a) r+\left(1+a+a^{2}\right) r^{2}+\left(1+a+a^{2}+a^{3}\right) r^{3}+\ldots \ldots ., r\) and \(a\) being proper fractions.

Find \(\lim _{x \rightarrow 1} \frac{x+x^{2}+\ldots \ldots . .+x^{n}-n}{x-1} .\)

Find \(\lim _{n \rightarrow \infty} n^{-n^{2}}\left((n+1)\left(n+\frac{1}{2}\right)\left(n+\frac{1}{2^{2}}\right) \cdots\left(n+\frac{1}{2^{n-1}}\right)\right)^{n}\)

Insert five geometric means between 486 and \(\frac{2}{3}\).

If \(A\) and \(G\) be the A.M. and G.M. between two numbers, prove that the numbers are \(A \pm \sqrt{(A+G)(A-G)}\).

Construct a quadratic in \(x\) such that A.M. of its roots is \(A\) and G.M. is \(G\).

If one G.M. \(G\) and two arithmetic means \(p\) and \(q\) be inserted between any two given numbers then show that \(G^{2}=(2 p-q)(2 q-p)\).

If one A.M. \(A\) and two geometric means \(p\) and \(q\) be inserted between any two given numbers then show that \(p^{3}+q^{3}=2\) Apq.

The A.M. between \(m\) and \(n\) and the G.M. between \(a\) and \(b\) are each equal to \((m a+n b)(m+n)\). Find \(m\) and \(n\) in terms of \(a\) and \(b\).

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

If \(A_{1}, A_{2}\) be two AM's and \(G_{1}, G_{2}\) be two GM's between \(a\) and \(b\), then prove that \(\frac{A_{1}+A_{2}}{G_{1} G_{2}}=\frac{a+b}{a b}\).

If the A.M. between \(a\) and \(b\) is twice as great as their G.M. show that \(a: b=(2+\sqrt{3}):(2-\sqrt{3})\).

The A.M. of \(a\) and \(b\) is to their G.M as \(m\) to \(n\), show \(a: b=m+\sqrt{m^{2}-n^{2}}: m-\sqrt{m^{2}-n^{2}}\).

If \(A\) be the arithmetic mean of \(b\) and \(c\) and \(G_{1}, G_{2}\), be the two geometric means between them, then prove that \(G_{1}^{3}+G_{2}^{3}=2 G_{1} G_{2} A\).

The 7th term of a H.P. is \(\frac{1}{10}\) and 12 th term is \(\frac{1}{25}\), find the 20 th term.

Solve the equation \(6 x^{3}-11 x^{2}+6 x-1=0\) if its roots are in harmonic progression.

The value of \(x+y+z\) is 15 if \(a, x, y, z, b\) are in A.P. while the value of \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\) is \(\frac{5}{3}\) if \(a, x, y, z, b\) are in H.P., find \(a\) and \(b\).

If \(x, y, z\) are in H.P., prove that \(\log (x+z)+\log (x+z-2 y)=2 \log (x-z)\).

Show that \(\log _{3} 2, \log _{6} 2, \log _{12} 2\) are in HP.

Prove that \(a, b, c\) are in A.P., G.P or H.P. according as the value of \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{a}, \frac{a}{b}\) or \(\frac{a}{c}\) respectively.

If \(a_{1}, a_{2}, a_{3}, \ldots ., a_{n}\) are in harmonic progression, prove that \(a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n-1} a_{n}=(n-1) a_{1} a_{n} .\)

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

If \(p\) th term of an H.P. is \(q r\) and \(q\) th term is \(r p\), prove that \(r\) th term is \(p q\).

If the roots of the equation \(a(b-c) x^{2}+b(c-a) x+c(a-b)=0\) be equal, then prove that \(a, b, c\) are in H.P.

If the \(m\) th term of an H.P. is \(n\) and \(n\) th term be \(m\), then prove that \((m+n)\) th term is \(\frac{m n}{m+n}\).

If \(a, b, c\) be in H.P., prove that i. \(\frac{a-b}{b-c}=\frac{a}{c}\); ii. \(\frac{1}{b-a}+\frac{1}{b-c}=\frac{2}{b}\); iii. \(\frac{b+a}{b-a}+\frac{b+c}{b-c}=2\).; iv. \(\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{b}+\frac{1}{c}-\frac{1}{a}\right)=\frac{4}{a c}-\frac{3}{b^{2}}\); v. \((b c+c a-a b)(c a+a b-b c)=a c\left(4 b^{2}-3 a c\right)\).

If \(a, b, c\) be in H.P. prove that i. \(\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}\) are in H.P.; ii. \(\frac{a}{b+c-a}, \frac{b}{c+a-b}, \frac{c}{a+b-c}\) are in H.P. iii. \(\frac{1}{a}+\frac{1}{b+c}, \frac{1}{b}+\frac{1}{c+a}, \frac{1}{c}+\frac{1}{a+b}\) are in H.P.

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

If \(b+c, c+a, a+b\) are in H.P. then prove that i. \(\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}\) are in A.P.; ii. \(a^{2}, b^{2}, c^{2}\) are in A.P.