Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
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 three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + 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 \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
Find the geometric means of the following pairs of number:
a3b and ab3
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
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 in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this 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
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
The third term of a G.P. is 4, the product of the first five terms is ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
