Advertisements
Advertisements
Question
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
Advertisements
Solution
Let the five numbers in G.P. be `"a"/"r"^2, "a"/"r","a", "ar", "ar"^2`
Since their product is 243,
`"a"/"r"^2."a"/r."a"."ar"."ar"^2`= 243
∴ a5 = 35
∴ a = 3
Also, the sum of second and fourth is 10
∴ `"a"/"r" + "ar"` = 10
∴ `3/"r" + 3"r"` = 10r ...[∵ a = 3]
∴ 3 + 3r2 = 10r
∴ 3r2 − 10r + 3 = 0
∴ (r – 3)(3r – 1) = 0
∴ r – 3 = 0 or 3r – 1= 0
∴ r = 3 or r = `1/3`
Taking r = `3, "a"/"r"^2 = 3/9 = 1/3, "a"/"r" = 3/3` = 1, ar = 3 × 3 = 9,
ar2 = 3(3)2 = 27 and the five numbers are `1/3`, 1, 3, 9, 27
Taking r = `1/3, "a"/"r"^2 = 3/((1/9)) = 27, "a"/"r" = 3/((1/3))` = 9,
ar = `3(1/3)` = 1, ar2 = `3(1/3)^2 = 1/3`
and the five numbers are 27, 9, 3, 1, `1/3`
Hence, the required numbers in G.P. are `1/3`, 1, 3, 9, 27 or 27, 9, 3, 1, `1/3`.
APPEARS IN
RELATED QUESTIONS
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Evaluate `sum_(k=1)^11 (2+3^k )`
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.
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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.
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:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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.
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.
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
Find the geometric means of the following pairs of number:
−8 and −2
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For the G.P. if r = − 3 and t6 = 1701, find a.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
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
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
