Advertisements
Advertisements
प्रश्न
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Advertisements
उत्तर
Sn = 4(7n – 1)
∴ Sn–1 = 4(7n–1 – 1)
But, tn = Sn – Sn–1
= 4(7n – 1) – 4(7n–1 – 1)
= 4(7n – 1 – 7n–1 + 1)
= 4(7n – 7n–1)
= 4(7n–1+1 – 7n–1)
= 4.7n–1 (7 – 1)
∴ tn = 24.7n–1
∴ tn–1 = `24.7^(("n" - 1) - 1)`
= 24.7n–2
The sequence is a G.P., if `"t"_"n"/"t"_("n" - 1)` = constant for all n ∈ N.
∴ `"t"_"n"/"t"_("n" - 1) = 24.7^("n" - 1)/24.7^("n" - 2) = 7^("n" - 1)/(7^("n" - 1).7^((-1))`
= 7
= constant, for all n ∈ N
∴ the given sequence is a G.P.
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
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`.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
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.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
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 S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
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 xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Find the geometric means of the following pairs of number:
2 and 8
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
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
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
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.
For a G.P. if S5 = 1023 , r = 4, Find a
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
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"`
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
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))`
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 ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
