Advertisements
Advertisements
Question
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Advertisements
Solution
Here, a = \[\frac{2}{9} \text { and }r = - \frac{3}{2}\] .
\[S_5 = a\left( \frac{r^5 - 1}{r - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\left( \frac{- 3}{2} \right)^5 - 1}{\frac{- 3}{2} - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\left( - \frac{243}{32} \right) - 1}{\frac{- 3}{2} - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\frac{- 275}{32}}{\frac{- 5}{2}} \right)\]
\[ = \frac{1100}{1440}\]
\[ = \frac{55}{72}\]
APPEARS IN
RELATED QUESTIONS
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
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
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.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
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.
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 = − 3 and t6 = 1701, find a.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 3 years.
The numbers x − 6, 2x and x2 are in G.P. Find x
For a G.P. a = 2, r = `-2/3`, find S6
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
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:
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:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
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 ______.
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
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 x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
