Advertisements
Advertisements
प्रश्न
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
उत्तर
0.6, 0.66, 0.666, 0.6666, …
∴ t1 = 0.6
t2 = 0.66 = 0.6 + 0.06
t3 = 0.666 = 0.6 + 0.06 + 0.006
Hence, in general
tn = 0.6 + 0.06 + 0.006 + … upto n terms.
The terms are in G.P. with
a = 0.6, r = `0.06/0.6` = 0.1
∴ tn = the sum of first n terms of the G.P.
∴ tn = `0.6[(1 - (0.1)^"n")/(1 - 0.1)] = 0.6/0.9[1 - (0.1)^"n"]`
∴ tn = `6/9[1 - (0.1)^"n"] = 2/3[1 - (0.1)^"n"]`
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
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
Find the sum of the following geometric series:
\[\sqrt{7}, \sqrt{21}, 3\sqrt{7}, . . .\text { to n terms }\]
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.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
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 numbers 3, x, and x + 6 form are in G.P. Find nth term
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. If t3 = 20 , t6 = 160 , find S7
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
