Advertisements
Advertisements
प्रश्न
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Advertisements
उत्तर
`2, 4/3, 8/9, 16/27, ...`
`"a" = 2, "r" = (4/3)/2 = 2/3, |"r"| < 1`
∴ Sum to infinity exists.
Sum to infinity = `"a"/(1 - "r")`
= `2/(1 - 2/3)`
= 6
APPEARS IN
संबंधित प्रश्न
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
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`.
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
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.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
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.
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, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
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 four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
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.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
The numbers x − 6, 2x and x2 are in G.P. Find nth term
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
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"`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
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 the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
