Advertisements
Advertisements
प्रश्न
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`.
उत्तर
Let the first term of the geometric progression be a and common ratio = `1/"r"^"n"`, then
Sum of n terms = `("a"(1 - "r"^"n"))/(1 - "r")` .....(i)
(n + 1)th term = `"ar"^("n"+ 1 - 1)` = arn
∴ arn + arn + 1 + arn + 2 + ....... up to n terms
= `("ar"^"n"(1 - "r"^"n"))/(1 - "r")` .....(ii)
Dividing equation (i) by (ii), we get
`("Sum of n terms")/("Sum of next n terms") = ("a"(1 - "r"^"n"))/(1 - "r") ÷ ("ar"^ "n"(1 - "r"^"n"))/(1 - "r")`
= `("a"(1 - "r"^"n"))/(1 - "r") xx (1 - "r")/("ar"^"n" (1 - "r"^ "n"))`
= `1/"r"^"n"`
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Evaluate `sum_(k=1)^11 (2+3^k )`
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
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 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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.
Express the recurring decimal 0.125125125 ... as a rational number.
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
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 the following is also in G.P.:
a2, b2, c2
If a, b, c are in G.P., then prove that:
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Find the geometric means of the following pairs of number:
2 and 8
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 the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
For the G.P. if a = `7/243`, r = 3 find t6.
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.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is –
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
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 ______.
