Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Advertisements
उत्तर
Common Ratio = r = `(-a)/1 = -a`
∴ Sum of GP for n terms = `(a(r^n - 1))/(r - 1)` ...(i)
⇒ a = 1, r = −a, n = n
∴ Substituting the above values in (1) we get
⇒ `(1((-a)^n - 1))/(-a-1)`
⇒ `(-1((-a)^n - 1))/(a+1)`
APPEARS IN
संबंधित प्रश्न
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
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.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.
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 rational number whose decimal expansion is `0.4bar23`.
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:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
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 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.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
Write the product of n geometric means between two numbers a and b.
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if a = `2/3`, t6 = 162, find r.
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
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 G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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 for a G.P. first term is (27)2 and seventh term is (8)2, find S8
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 P2 R3 : S3 is equal to ______.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
