Advertisements
Advertisements
Question
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
Advertisements
Solution
k, k + 9, k−6 are in G.P.
\[\therefore \left( k - 6 \right)^2 = 4\left( k + 9 \right)\]
\[ \Rightarrow k^2 + 36 - 12k = 4k + 36\]
\[ \Rightarrow k^2 - 16k = 0\]
\[ \Rightarrow k \left( k - 16 \right) = 0\]
\[ \Rightarrow k = 0, 16\]
But, k = 0 is not possible.
∴ k = 16
APPEARS IN
RELATED QUESTIONS
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
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:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
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.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
2 and 8
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
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
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
