Find the \(20^{\text{th}}\) and \(n^{\text{th}}\) terms of the G.P. \( \dfrac{5}{2}, \dfrac{5}{4}, \dfrac{5}{8}, \ldots \).
\(a_{20} = \dfrac{5}{2^{20}},\ a_n = \dfrac{5}{2^n}\).
Find the \(12^{\text{th}}\) term of a G.P. whose \(8^{\text{th}}\) term is \(192\) and the common ratio is \(2\).
\(a_{12} = 3072\).
The \(5^{\text{th}}\), \(8^{\text{th}}\) and \(11^{\text{th}}\) terms of a G.P. are \(p\), \(q\) and \(s\), respectively. Show that \(q^{2} = ps\).
The \(4^{\text{th}}\) term of a G.P. is the square of its second term, and the first term is \(-3\). Determine its \(7^{\text{th}}\) term.
The \(7^{\text{th}}\) term is \(-2187\).
Which term of each of the following sequences has the indicated value?
(a) In the sequence \(2, 2\sqrt{2}, 4, \ldots\), which term is \(128\)?
(b) In the sequence \(\sqrt{3}, 3, 3\sqrt{3}, \ldots\), which term is \(729\)?
(c) In the sequence \( \dfrac{1}{3}, \dfrac{1}{9}, \dfrac{1}{27}, \ldots \), which term is \( \dfrac{1}{19683} \)?
(a) \(13^{\text{th}}\) term
(b) \(12^{\text{th}}\) term
(c) \(9^{\text{th}}\) term
For what values of \(x\) are the numbers \(-\dfrac{2}{7},\ x,\ \dfrac{7}{2}\) in G.P.?
\(x = \pm 1\).
Find the sum to the indicated number of terms in the geometric progression.
\(0.15,\ 0.015,\ 0.0015, \ldots\) \(20\) terms.
\(S_{20} = \dfrac{1}{6}\big(1 - (0.1)^{20}\big)\).
Find the sum to \(n\) terms of the geometric progression.
\(\sqrt{7},\ \sqrt{21},\ 3\sqrt{7}, \ldots,\ n\text{ terms}.\)
\(S_n = \dfrac{\sqrt{7}}{2}(\sqrt{3}+1)\big(3^{\dfrac{n}{2}} - 1\big)\).
Find the sum to \(n\) terms of the geometric progression \(1, -a, a^{2}, -a^{3}, \ldots\) (if \(a \neq -1\)).
\(S_n = \dfrac{1 - (-a)^{n}}{1 + a}\).
Find the sum to \(n\) terms of the geometric progression \(x^{3}, x^{5}, x^{7}, \ldots\) (if \(x \neq \pm 1\)).
\(S_n = \dfrac{x^{3}\big(1 - x^{2n}\big)}{1 - x^{2}}\).
Evaluate \(\displaystyle \sum_{k=1}^{11} (2 + 3^{k})\).
\(\displaystyle \sum_{k=1}^{11} (2 + 3^{k}) = 22 + \dfrac{3}{2}\big(3^{11} - 1\big)\).
The sum of the first three terms of a G.P. is \(\dfrac{39}{10}\) and their product is \(1\). Find the common ratio and the three terms.
Common ratio \(r = \dfrac{5}{2}\) or \(r = \dfrac{2}{5}\).
The three terms are \(\dfrac{2}{5}, 1, \dfrac{5}{2}\) or \(\dfrac{5}{2}, 1, \dfrac{2}{5}\).
How many terms of the G.P. \(3, 3^{2}, 3^{3}, \ldots\) are needed to give the sum \(120\)?
\(4\) terms.
The sum of the first three terms of a G.P. is \(16\) and the sum of the next three terms is \(128\). Determine the first term, the common ratio and the sum to \(n\) terms of the G.P.
First term \(a = \dfrac{16}{7}\).
Common ratio \(r = 2\).
Sum to \(n\) terms: \(S_n = \dfrac{16}{7}\big(2^{n} - 1\big)\).
Given a G.P. with \(a = 729\) and \(7^{\text{th}}\) term \(64\), determine \(S_{7}\).
\(S_{7} = 2059\) or \(S_{7} = 463\).
Find a G.P. for which the sum of the first two terms is \(-4\) and the fifth term is four times the third term.
One possible G.P. is \( -\dfrac{4}{3}, -\dfrac{8}{3}, -\dfrac{16}{3}, \ldots \)
Another possible G.P. is \( 4, -8, 16, -32, 64, \ldots \)
If the 4th, 10th and 16th terms of a G.P. are \(x\), \(y\) and \(z\), respectively, prove that \(x\), \(y\), \(z\) are in G.P.
Find the sum to \(n\) terms of the sequence \(8, 88, 888, 8888, \ldots\).
\( \dfrac{80}{81}\bigl(10^{n} - 1\bigr) - \dfrac{8}{9}n \)
Find the sum of the products of the corresponding terms of the sequences \(2, 4, 8, 16, 32\) and \(128, 32, 8, 2, \tfrac{1}{2}\).
\(496\)
Show that the products of the corresponding terms of the sequences \(a, ar, ar^{2}, \ldots, ar^{n-1}\) and \(A, AR, AR^{2}, \ldots, AR^{n-1}\) form a G.P., and find the common ratio.
The common ratio of the G.P. is \(rR\).
Find four numbers forming a geometric progression in which the third term is greater than the first term by \(9\), and the second term is greater than the fourth term by \(18\).
The four numbers are \(3, -6, 12, -24\).
If the \(p^{\text{th}}\), \(q^{\text{th}}\) and \(r^{\text{th}}\) terms of a G.P. are \(a\), \(b\) and \(c\), respectively, prove that \(a^{\,q-r} b^{\,r-p} c^{\,p-q} = 1\).
If the first and the \(n^{\text{th}}\) term of a G.P. are \(a\) and \(b\), respectively, and if \(P\) is the product of \(n\) terms, prove that \(P^{2} = (ab)^{n}\).
Show that the ratio of the sum of the first \(n\) terms of a G.P. to the sum of the terms from \((n+1)^{\text{th}}\) to \((2n)^{\text{th}}\) term is \(\dfrac{1}{r^{n}}\), where \(r\) is the common ratio.
If \(a, b, c\) and \(d\) are in G.P., show that
\((a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d^{2}) = (ab + bc + cd)^{2}.\)
Insert two numbers between \(3\) and \(81\) so that the resulting sequence is a G.P.
The two numbers are \(9\) and \(27\).
Find the value of \(n\) so that \( \dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}} \) may be the geometric mean between \(a\) and \(b\).
\(n = -\dfrac{1}{2}\)
The sum of two numbers is six times their geometric mean. Show that the numbers are in the ratio \((3 + 2\sqrt{2}) : (3 - 2\sqrt{2})\).
If \(A\) and \(G\) be A.M. and G.M., respectively, between two positive numbers, prove that the numbers are \(A \pm \sqrt{(A+G)(A-G)}\).
The number of bacteria in a certain culture doubles every hour. If there were \(30\) bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and \(n^{\text{th}}\) hour?
After 2nd hour: \(120\); after 4th hour: \(480\); after \(n^{\text{th}}\) hour: \(30 \cdot 2^{n}\).
What will Rs \(500\) amount to in 10 years after its deposit in a bank which pays an annual interest rate of \(10\%\) compounded annually?
Rs \(500 (1.1)^{10}\)
If A.M. and G.M. of the roots of a quadratic equation are \(8\) and \(5\), respectively, obtain the quadratic equation.
\(x^{2} - 16x + 25 = 0\)