Advertisements
Advertisements
प्रश्न
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
Advertisements
उत्तर
tn = `(5^("n"-3))/(2^("n"-3)) = (5/2)^("n"-3)`
∴ tn+1 = `(5/2)^("n"+1-3) = (5/2)^("n"-2)`
∴ `("t"_("n"+1))/"t"_"n" = ((5/2)^("n"-2))/((5/2)^("n"-3))`
= `(5/2)^("n" - 2 - "n" + 3)`
= `5/2`, which is a constant
∴ the sequence is a G.P. whose common ratio is `5/2`.
Now, tn = `(5/2)^("n" - 3)`
∴ the first term = t1 = `(5/2)^(1 - 3)`
= `(5/2)^(-2)`
= `(2/5)^2`
= `4/25`
Hence, the first term = t1 = `4/25`
and the common ratio = r = `5/2`.
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
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`.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is `1/r^n`.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
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
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
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.
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.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Mark the correct alternative in the following question:
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 p2R3 : S3 is equal to
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For the G.P. if a = `7/243`, r = 3 find t6.
Which term of the G.P. 5, 25, 125, 625, … is 510?
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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]
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
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
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` 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 ______.
