Advertisements
Advertisements
प्रश्न
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Advertisements
उत्तर
Given series has sum S = `1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
S = `(1/3+1/3^3+1/3^5...infty)+(1/5^2+1/5^4+1/5^6...infty)`
We know sum of a G.P. up to infinity is given by, S = `a/(1 – r)`.
Let S1 = `1/3+1/3^3+1/3^5...infty`
This is an infinite G.P. with first term(a) = 1/3 and common ratio(r) = 1/32 = 1/9.
So, S1 = `(1/3)/(1-1/9)`
= `3/8`
Let S2 = `1/5^2+1/5^4+1/5^6...infty`
This is an infinite G.P. with first term (a) = `1/5^2` and common ratio (r) = `1/5^2 = 1/25`.
So, S2 = `(1/25)/(1-1/25)`
= `1/24`
Now, required sum, S = S1 + S2
= `3/8+1/24`
= `10/24`
= `5/12`
Therefore, sum of the series to infinity is `5/12`.
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.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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 four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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`.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
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.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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 a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
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.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
The two geometric means between the numbers 1 and 64 are
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in 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]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`0.bar(7)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
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:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
