Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text{ Let r be the common ratio of the given G . P } . \]
\[ \therefore a = 4\]
\[\text { Sum of the geometric ifinite series: } \]
\[ S_\infty = 4 + 4r + 4 r^2 + . . . \infty \]
\[\text { Now, } S_\infty = \frac{4}{1 - r} . . . . . . . \left( i \right)\]
\[\text { The difference between the third and fifth term is } \frac{32}{81} . \]
\[ a_3 - a_5 = \frac{32}{81}\]
\[ \Rightarrow 4 r^2 - 4 r^4 = \frac{32}{81}\]
\[ \Rightarrow 4\left( r^2 - r^4 \right) = \frac{32}{81}\]
\[ \Rightarrow 81 r^4 - 81 r^2 + 8 = 0 . . . . . . . \left( ii \right)\]
\[\text { Now, let } r^2 = y\]
\[\text { Let us put this in } \left( ii \right) . \]
\[ \therefore 81 r^4 - 81 r^2 + 8 = 0\]
\[ \Rightarrow 81 y^2 - 81y + 8 = 0\]
\[ \Rightarrow 81 y^2 - 72y - 9y + 8 = 0\]
\[ \Rightarrow 9y\left( 9y - 1 \right) - 8\left( 9y - 1 \right) = 0\]
\[ \Rightarrow \left( 9y - 8 \right)\left( 9y - 1 \right)\]
\[ \Rightarrow y = \frac{1}{9}, \frac{8}{9}\]
\[\text { Putting y } = r^2 ,\text { we get } r = \frac{1}{3} \text { and } \frac{2\sqrt{2}}{3}\]
\[\text { Substituting r } = \frac{1}{3} \text { and }\frac{2\sqrt{2}}{3} \text { in } \left( i \right): \]
\[ S_\infty = \frac{4}{1 - \frac{1}{3}} = \frac{12}{2} = 6\]
\[\text { Similarly }, S_\infty = \frac{4}{1 - \frac{2\sqrt{2}}{3}} = \frac{12}{3 - 2\sqrt{2}}\]
\[ \therefore S_\infty = 6, \frac{12}{3 - 2\sqrt{2}}\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
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:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
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.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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.
Find the geometric means of the following pairs of number:
2 and 8
If logxa, ax/2 and logb x are in G.P., then write the value of x.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
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
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
For the G.P. if a = `7/243`, r = 3 find t6.
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 n years.
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
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
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:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
