Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
tn = `(5^("n" - 2))/(7^("n" - 3)) = (5.5^("n" - 3))/(7^("n" - 3))`
∴ tn = `5(5/7)^("n" - 3)`
∴ tn+1 = `5(5/7)^("n" + 1 - 3)`
= `5(5/7)^("n" - 2)`
∴ `("t"_("n" + 1))/"t"_"n" = (5(5/7)^("n" - 2))/(5(5/7)^("n" - 3))`
= `(5/7)^("n" - 2 - "n" + 3)`
= `5/7`, which is a constant
∴ the sequence is a G.P. whose common ratio is `5/7`
Now, tn = `5(5/7)^("n" - 3)`
∴ the first term = t1 = `5(5/7)^(1 - 3)`
= `5(5/7)^(-2)`
= `5(7/5)^2`
= `49/5`
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 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:
\[\sum^{10}_{n = 2} 4^n\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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).
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.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
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 \[\frac{A_1 + A_2}{G_1 G_2}\]
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 product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
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.
For the following G.P.s, find Sn
3, 6, 12, 24, ...
For a G.P. If t4 = 16, t9 = 512, find S10
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
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
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
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:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
