Advertisements
Advertisements
Question
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Solution
Let the three numbers in G.P. be `"a"/"r", "a", "ar"`.
According to the given conditions,
`"a"/"r" + "a" + "ar"` = 35
∴ `"a"(1/"r" + 1 + "r")` = 35 ....(i)
Also, `("a"/"r")("a")("ar")` = 1000
∴ a3 = 1000
∴ a = 10
Substituting the value of a in (i), we get
`10(1/"r" + 1 + "r")` = 35
∴ `1/"r" + "r" + 1 = 35/10`
∴ `1/"r" + "r" = 35/10 - 1`
∴ `1/"r" + "r" = 25/10`
∴ `1/"r" + "r" = 5/2`
∴ 2r2 – 5r + 2 = 0
∴ (2r – 1) (r – 2) = 0
∴ r = `1/2` or r = 2
When r = `1/2`, a = 10
`"a"/"r" = 10/((1/2))` = 20, a = 10 and ar = `10(1/2)` = 5
When r = 2, a = 10
`"a"/"r" = 10/2` = 5, a = 10 and ar = 10 (2) = 20
Hence, the three numbers in G.P. are 20, 10, 5 or 5, 10, 20.
APPEARS IN
RELATED QUESTIONS
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
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.
How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following serie to infinity:
\[\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{56} + . . . \infty\]
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the sum of the terms of an infinite 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 32/81.
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 are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
The two geometric means between the numbers 1 and 64 are
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
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.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn
3, 6, 12, 24, ...
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
