Advertisements
Advertisements
प्रश्न
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Advertisements
उत्तर
`sum_("r" = 1)^oo 4(0.5)^"r" = 4 sum_("r" = 1)^oo (0.5)^"r"`
= 4[0.5 + (0.5)2 + (0.5)3 + ...]
= `4[5/10 + (5/10)^2 + (5/10)^3 + ...]`
= `4[1/2 + (1/2)^2 + (1/2)^3 + ....]` ...(1)
The terms `1/2, (1/2)^2, (1/2)^3` ... form a G.P. with
a = `1/2`, r = `1/2`
Since |r| = `|1/2| = 1/2 < 1`, the sum to infinity of this G.P. exist and
S = `"a"/(1 - "r")`
= `((1/2))/(1 - (1/2))` = 1
∴ from (1),
`sum_("r" = 1)^oo 4(0.5)^"r"` = 4 x 1 = 4.
APPEARS IN
संबंधित प्रश्न
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Given a G.P. with a = 729 and 7th term 64, determine S7.
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
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
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:
a (b2 + c2) = c (a2 + b2)
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
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 A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
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 S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If second term of a G.P. is 2 and the sum of its infinite terms is 8, 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 is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
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
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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 -
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
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 ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
