Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
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.
Express the recurring decimal 0.125125125 ... as a rational number.
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:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
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 logxa, ax/2 and logb x are in G.P., then write the value of x.
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.
The fractional value of 2.357 is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
The numbers 3, x, and x + 6 form are in G.P. Find nth term
For the following G.P.s, find Sn
3, 6, 12, 24, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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 in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
