Advertisements
Advertisements
Question
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Advertisements
Solution
\[\text { Let the 6 G . M . s between 27 and } \frac{1}{81}\text { be} G_1 , G_2 , G_3 , G_4 , G_5 \text { and } G_6 . \]
\[\text { Thus }, 27, G_1 , G_2 , G_3 , G_4 , G_5 , G_6 \text { and } \frac{1}{81} \text { are in G . P } . \]
\[ \therefore a = 27, n = 8 \text { and } a_8 = \frac{1}{81}\]
\[ \because a_8 = \frac{1}{81}\]
\[ \Rightarrow {ar}^7 = \frac{1}{81}\]
\[ \Rightarrow r^7 = \frac{1}{81 \times 27}\]
\[ \Rightarrow r^7 = \left( \frac{1}{3} \right)^7 \]
\[ \Rightarrow r = \frac{1}{3}\]
\[ \therefore G_1 = a_2 = ar = 27\left( \frac{1}{3} \right) = 9\]
\[ G_2 = a_3 = a r^2 = 27 \left( \frac{1}{3} \right)^2 = 3\]
\[ G_3 = a_4 = a r^3 = 27 \left( \frac{1}{3} \right)^3 = 1\]
\[ G_4 = a_5 = a r^4 = 27 \left( \frac{1}{3} \right)^4 = \frac{1}{3}\]
\[ G_5 = a_6 = a r^5 = 27 \left( \frac{1}{3} \right)^5 = \frac{1}{9} \]
\[ G_6 = a_7 = a r^6 = 27 \left( \frac{1}{3} \right)^6 = \frac{1}{27}\]
APPEARS IN
RELATED QUESTIONS
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
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.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
For the G.P. if r = − 3 and t6 = 1701, find a.
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 3 years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
For a G.P. If t3 = 20 , t6 = 160 , find S7
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
