Advertisements
Advertisements
प्रश्न
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Advertisements
उत्तर
Here, a = – 3, r = `-1/3`
Since | r | = `|-1/3| = 1/3 < 1`, the sum to infinity of this G.P. exist and
S = `"a"/(1 - "r")`
= `(-3)/(1 - (-1/3))`
= `(-3)/((4/3))`
= `-9/4`.
APPEARS IN
संबंधित प्रश्न
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`
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
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
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
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.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
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.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., then prove that:
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.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
For the G.P. if r = − 3 and t6 = 1701, find a.
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
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]
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" = 1)^oo 4(0.5)^"r"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
The third term of a G.P. is 4, the product of the first five terms is ______.
