Advertisements
Advertisements
Question
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Advertisements
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 indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
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.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 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.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
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 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
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if r = − 3 and t6 = 1701, find a.
Which term of the G.P. 5, 25, 125, 625, … is 510?
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
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.
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
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"`
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
