Chapter 9

Find the three numbers constituting a G.P. if it is known that the sum of the numbers is equal to 26 and that when 1,6 and 3 are added to them respectively, the new numbers are obtained which from an A.P.

Three numbers from a G.P. If the 3rd term is decreased by 64 , then the numbers thus obtained will constitute an A.P. If the second term of this A.P. is decreased by 8 , a G.P. will be formed again. Determine the numbers.

Find the numbers \(a, b, c\) between 2 and 18 such that (i) their sum is 25 (ii) the numbers \(2, a, b\) are consecutive terms of an A.P and (iii) the numbers \(b, c, 18\) are consecutive terms of a G.P.

In a G.P. if the \((m+n)\) th term be \(p\) and \((m-n)\) th term be \(q\), then prove that its \(m\) th term is \(\sqrt{p q}\).

The first and second terms of a GP are \(x^{-4}\) and \(x^{n}\) respectively. If \(x^{52}\) is the eight term of the same progression, then find \(n\).

If \(p, q, r\) are in AP, then show that \(p t h, q t h\) and \(r t h\) terms of any GP are in GP.

If \(a, b, c\) are in AP, \(b-a, c-b\) and \(a\) are in GP, then find \(a: b: c\).

If the roots of \(\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)\) are equal then show that \(a, b, c\) are in GP.

Let \(\left\\{a_{n}\right\\}\) be a GP such that \(\frac{a_{4}}{a_{6}}=\frac{1}{4}\) and \(a_{2}+a_{5}=216\). Then find \(a_{1}\).

The \(r\) th, \(s\) th and \(t\) th terms of a certain G.P. are \(R, S\) and \(T\) respectively. Prove that \(R^{s-t} \cdot S^{t-r} \cdot T^{r-s}=1\).

The fourth, seventh and tenth terms of a GP are \(p, q, r\) respectively, then show that \(q^{2}=p r\).

If \(x, y, z\) be respectively the \(p t h, q\) th and \(r\) th terms of a G.P., then prove that \((q-r) \log x+(r-p) \log y+(p-q) \log z=0\)

If the \(p\) th, \(q\) th, \(r\) th terms of an A.P. are in G.P. show that common ratio of the G.P. is \(\frac{q-r}{p-q}\).

If \(a, b, c, d\) be in G.P., prove that i. \(\left(a^{2}+a c+c^{2}\right)\left(b^{2}+b d+d^{2}\right)=(a b+b c+c d)^{2}\); ii. \(\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2}\).

If \(a, b, c\) be in G.P., then prove that \(\frac{a^{2}+a b+b^{2}}{b c+c a+a b}=\frac{b+a}{c+b}\).

If \(a, b, c\) are three distinct real numbers in G.P. and \(a+b+c=x b\), then prove that either \(x<-1\) or \(x>3\).

Find all the numbers \(x\) and \(y\) and such that \(x, x+2 y, 2 x+y\) from an A.P. while the numbers \((y+1)^{2}\), \(x y+5,(x+1)^{2}\) form a G.P. Write down the progressions.

If \(a, b, c, x\) are all real numbers, and \(\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0\) then show that \(a, b, c\) are in G.P., and \(x\) is their common ratio.

If \(a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}\) and \(a, b, c\) be in G.P. then prove that \(x, y, z\) are in A.P.

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

If the \(m\) th, \(n\) th, and \(p\) th terms of an A.P. and G.P. be equal and be respectively \(x, y\) and \(z\), then prove that \(x^{y-z} \cdot y^{z-x} \cdot z^{x-y}=1\)

If \(a, b, c\) are in G.P. and \(x, y\) respectively be arithmetic means between \(a, b\) and \(b, c\), then prove that \(\frac{a}{x}+\frac{c}{y}=2\) and \(\frac{1}{x}+\frac{1}{y}=\frac{2}{b}\)

If \(a, b, c\) be distinct positive and in G.P. and \(\log _{c} a, \log _{b} c, \log _{a} b\) be in A.P. then show that the common difference of this A.P. is \(\frac{3}{2}\).

Prove that the three successive terms of a G.P. will form the sides of a triangle if the common ratio \(r\) satisfies the inequality \(\frac{1}{2}(\sqrt{5}-1)

The fifth term of a G.P. is 81 whereas its second term is 24 . Find the series and sum of its first eight terms.

The sum of first three of a G.P. is to the sum of the first six terms as \(125: 152\). Find the common ratio of the G.P.

For a sequence \(\left\\{a_{n}\right\\}, a_{1}=2\) and \(\frac{a_{n+1}}{a_{n}}=\frac{1}{3}\). Then find the value of \(\sum_{r=1}^{20} a_{r}\).

Find the value of \(9^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 9^{\frac{1}{27}} \ldots \ldots \ldots .\) upto \(\infty\).

Find the value of \(x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{8}} \cdot x^{\frac{1}{16}} \ldots \ldots\)

If \(S=1+2+4+8+16+32+\ldots \ldots \infty\), then \(S\) is a positive number. Multiply both sides by 2 , then it is found that \(2 S=S-1\) which leads to conclusion \(S=-1\) which is certainly negative. Do you agree with the conclusion? Your answer should be supported by explanation.

The first term of an infinite G.P. is 1 and any term is equal to the sum of all the succeeding terms. Find the series.

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of succeeding terms. Find the series.

Sum of a certain number of terms of the series \(\frac{2}{9}-\frac{1}{3}+\frac{1}{2}-\cdots \cdots\) is \(\frac{55}{72}\). Find the number.

How many terms of the series \(1,4,16, \ldots\) must be taken to have their sum equal to 341 ?

In a G.P. sum of \(n\) terms is 255 , the last term is 128 and common ratio is 2 . Find \(n\).

In an increasing G.P., the sum of the first and the last term is 66, the product of the second and the last but one term is 128 , and the sum of all the terms is 126 . How many terms are there in the progression?

In a G.P. sum of \(n\) terms is 364 , first term is 1 and the common ratio is 3 . Find \(n\).

Express the recurring decimal \(0.125125125 \ldots \ldots\) as a rational number.

Find the value of \(0.1 \overline{23}\) regarding it as a geometric series.

Find the value of \(0.4 \overline{23}\).

Find the value of \(2 . \overline{357}\)

After striking a floor a certain ball rebounds \(\frac{4}{5}\) th of the height from which it has fallen. Find the total distance that it travels before coming to rest, if it is gently dropped from a height of 120 meters.

A ball is dropped from a height of \(48 \mathrm{ft}\). and rebounds two-third of the distance it falls. If it continues to fall and rebound in this way, how far will it travel before coming to rest?

A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then find the common ratio.

If in an infinite GP, first term is equal to 10 times the sum of all successive terms, then find its common

If second term of a GP is 2 and the sum of its infinite terms is 8 , then find its first term.

Find the sum of the term of an infinitely decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to \(\frac{32}{81}\).

The sum of an infinite geometric progression is 2 and the sum of the geometric progression made from the cube of the terms of this infinite series is 24 . Then find the series.

If the sum of an infinite GP be 3 and the sum of squares of its term is also 3 , then find its first term and common ratio.

The length of a side of a square is \(a\) meters. A second square is formed by joining the middle points of this square. Then a third square is formed by joining the middle points of the second square and so on. The process is carried on ad-infinitum. Find the sum of the areas of the squares.