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
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Given a G.P. with a = 729 and 7th term 64, determine S7.
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Find three numbers in G.P. whose sum is 38 and their product is 1728.
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.
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
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 \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
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 a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
a3b and ab3
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
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
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For the G.P. if r = − 3 and t6 = 1701, find a.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
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 p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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 of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
