Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Advertisements
उत्तर
Here, a = 0.15 and r \[= \frac{a_2}{a_1} = \frac{0 . 015}{0 . 15} = \frac{1}{10}\] .
\[S_8 = a\left( \frac{1 - r^8}{1 - r} \right) \]
\[ = 0 . 15\left( \frac{1 - \left( \frac{1}{10} \right)^8}{1 - \frac{1}{10}} \right)\]
\[ = 0 . 15\left( \frac{1 - \frac{1}{{10}^8}}{\frac{1}{10}} \right)\]
\[ = \frac{1}{6}\left( 1 - \frac{1}{{10}^8} \right)\]
संबंधित प्रश्न
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
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:
−2/3, −6, −54, ...
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
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 series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Express the recurring decimal 0.125125125 ... as a rational number.
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.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
Write the product of n geometric means between two numbers a and b.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
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
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
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.
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 10 years.
For a G.P. If t3 = 20 , t6 = 160 , find S7
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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,...`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
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 areas of all the squares
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
