Advertisements
Advertisements
Question
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Advertisements
Solution
We have,
\[\frac{a_2}{a_1} = \frac{0 . 02}{0 . 004} = 5, \frac{a_3}{a_2} = \frac{0 . 1}{0 . 02} = 5\]
\[ \Rightarrow \frac{a_2}{a_1} = \frac{a_3}{a_2} = 5\]
\[\text { The given progression is a G . P . whose first term, a is 0 . 004 and common ratio, r is 5 }. \]
\[\text { Let the nth term be } 12 . 5 . \]
\[ \therefore a_n = 12 . 5\]
\[ \Rightarrow a r^{n - 1} = 12 . 5\]
\[ \Rightarrow (0 . 004)(5 )^{n - 1} = 12 . 5\]
\[ \Rightarrow (5 )^{n - 1} = \frac{12 . 5}{0 . 004}\]
\[ \Rightarrow (5 )^{n - 1} = 3125\]
\[ \Rightarrow (5 )^{n - 1} = (5 )^5 \]
\[\text { Comparing the power of both the sides }\]
\[ \Rightarrow n - 1 = 5\]
\[ \Rightarrow n = 6\]
\[\text { Thus, 6th term of the given G . P . is } 12 . 5\]
RELATED QUESTIONS
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
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 8th term of the G.P. 0.3, 0.06, 0.012, ...
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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).
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.
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: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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 in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
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:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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.:
a2, b2, c2
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
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 a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
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 for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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
The sum or difference of two G.P.s, is again a G.P.
