Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Advertisements
उत्तर
`a/(1 + i) + a/(1 + i)^2 + a/(1 + i)^3 + ...... + a/(1 + i)^n`
∴ First term, A = `a/(1 + i)`, No. of terms = n,
Common ratio, R = `(a/(1 + i)^2)/(a/(1 + i))`
R = `(cancel(a)/cancel((1 + i))^2)/(cancel(a)/cancel(1 + i))`
∴ R = `1/(1 + i)`
`"S"_"n" = "A" [(1 - "R"^n)/(1 - "R")]`
`= a/(1 + i) [(1 - (1/(1 + i))^n)/(1 - 1/(1 + i))]`
`= a/cancel(1 + i) [(1 - 1/(1 + i)^n)/((cancel(1) + i - cancel(1))/cancel(1 + i))]`
`= a/i xx i/i [1 - (1 + i)^-n]`
`= (ai)/i^2 [1 - (1 + i)^-n]`
`= (ai)/-1 [1 - (1 + i)^-n]`
= - ai [1 - (1 + i)-n]
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.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
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 \[\frac{1}{r^n}\].
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, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in G.P., then prove that:
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x 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 = `1/3`, a = 9 find t7
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 ______.
The sum or difference of two G.P.s, is again a 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.
