Advertisements
Advertisements
प्रश्न
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
Advertisements
उत्तर
Let a be the first term and r be the common ratio of the G.P. According to the given conditions,
S2 = `-4 = ("a"(1 - "r"^2))/(1 - "r")` ........(i)
a5 = 4 × a3
⇒ ar4 = 4ar2 ⇒ r2 = 4
∴ r = ± 2
From (i) we obtain
-4 = `("a"[1 - (2)^2])/(1 - 2)` for r = 2
⇒ `-4 = ("a"(1 - 4))/-1`
⇒ −4 = a(3)
⇒ a = `(-4)/3`
Also, −4 = `("a"[1 - (-2)^2])/(1 - (-2))` for r = −2
⇒ `-4 = ("a"(1 - 4))/(1 + 2)`
⇒ `-4 = ("a"(-3))/3`
⇒ a = 4
Thus, the required G. P. is `(-4)/3, (-8)/3, (-16)/3` ,.... or 4, −8, 16, −32 ........
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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.
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
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find the 4th term from the end of the G.P.
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
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 series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
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`
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
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 are in G.P., then prove that:
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.
Find the geometric means of the following pairs of number:
2 and 8
Find the geometric means of the following pairs of number:
a3b and ab3
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
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 two geometric means between the numbers 1 and 64 are
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. If t4 = 16, t9 = 512, find S10
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
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 areas of all the squares
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n 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 ______.
