Advertisements
Advertisements
प्रश्न
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
Advertisements
उत्तर
Here, a = 3
Common ratio,
\[r = \frac{1}{2}\]
Sn = \[\frac{3069}{512}\]
\[\therefore S_n = 3\left\{ \frac{1 - \left( \frac{1}{2} \right)^n}{1 - \frac{1}{2}} \right\}\]
\[ \Rightarrow \frac{3069}{512} = 3\left\{ \frac{1 - \frac{1}{2^n}}{\frac{1}{2}} \right\} \]
\[ \Rightarrow \frac{3069}{512} = 6 \left\{ 1 - \frac{1}{2^n} \right\}\]
\[ \Rightarrow \frac{3069}{3072} = 1 - \frac{1}{2^n} \]
\[ \Rightarrow \frac{1}{2^n} = 1 - \frac{3069}{3072} \]
\[ \Rightarrow \frac{1}{2^n} = \frac{3}{3072}\]
\[ \Rightarrow 2^n = \frac{3072}{3}\]
\[ \Rightarrow 2^n = 1024 \]
\[ \Rightarrow 2^n = 2^{10} \]
\[ \therefore n = 10\]
APPEARS IN
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Given a G.P. with a = 729 and 7th term 64, determine S7.
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
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.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
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.
Write the product of n geometric means between two numbers a and b.
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
The numbers x − 6, 2x and x2 are in G.P. Find x
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
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 ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
