Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Advertisements
उत्तर
Here,a = 4 and r = \[\frac{1}{2}\]
\[\therefore S_n = a\left( \frac{1 - r^{10}}{1 - r} \right)\]
\[ = 4\left( \frac{1 - \left( \frac{1}{2} \right)^{10}}{1 - \left( \frac{1}{2} \right)} \right)\]
\[ = 4\left( \frac{1 - \left( \frac{1}{1024} \right)}{\frac{1}{2}} \right)\]
\[ = 8\left( 1 - \frac{1}{1024} \right)\]
\[ = \frac{1023}{128}\]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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 an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c are in G.P., prove that log a, log b, log c 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 A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
Write the product of n geometric means between two numbers a and b.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
For the G.P. if a = `7/243`, r = 3 find t6.
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
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 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 ______.
The sum or difference of two G.P.s, is again a G.P.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
