Advertisements
Advertisements
प्रश्न
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Advertisements
उत्तर
\[\text { Given }: \]
\[\text { First term, } a = 5 \]
\[\text { Common ratio }, r = 2\]
\[ a_n = \left( 5 \right) \left( 2 \right)^{n - 1} . . . \left( 1 \right)\]
\[\text { Similarly, } a_n = \left( 1280 \right) \left( \frac{1}{2} \right)^{n - 1} . . . \left( 2 \right)\]
\[\text { From }\left( 1 \right) \text { and } \left( 2 \right)\]
\[\left( 5 \right) \left( 2 \right)^{n - 1} = \left( 1280 \right) \left( \frac{1}{2} \right)^{n - 1} \]
\[ \Rightarrow \frac{1}{256} = \left( \frac{1}{4} \right)^{n - 1} \]
\[ \Rightarrow \left( \frac{1}{4} \right)^4 = \left( \frac{1}{4} \right)^{n - 1} \]
\[ \Rightarrow n - 1 = 4 \]
\[ \Rightarrow n = 5\]
संबंधित प्रश्न
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… .
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
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.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The fractional value of 2.357 is
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
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 a = `7/243`, r = 3 find t6.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
For a G.P. a = 2, r = `-2/3`, find S6
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
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.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... 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 -
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
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 ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
