हिंदी

For the following G.P.s, find Sn 0.7, 0.07, 0.007, .....

Advertisements
Advertisements

प्रश्न

For the following G.P.s, find Sn

0.7, 0.07, 0.007, .....

योग
Advertisements

उत्तर

a = 0.7 = `7/10`, r = `0.07/0.7 = 1/10 < 1`

Sn = `("a"(1 - "r"^"n"))/(1 - "r")`, for r < 1

= `(7/10[1 - (1/10)^"n"])/(1 - 1/10)`

= `7/9 (1 - 1/10^"n")`

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 2: Sequences and Series - Exercise 2.2 [पृष्ठ ३१]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 11 Maharashtra State Board
अध्याय 2 Sequences and Series
Exercise 2.2 | Q 1. (iii) | पृष्ठ ३१

संबंधित प्रश्न

Which term of the following sequence:

`1/3, 1/9, 1/27`, ...., is `1/19683`?


Find the sum to n terms of the sequence, 8, 88, 888, 8888… .


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 `1/r^n`.


The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.


Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.


Find :

the 8th term of the G.P. 0.3, 0.06, 0.012, ...


The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.


In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.


Find three numbers in G.P. whose sum is 38 and their product is 1728.


The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.


Evaluate the following:

\[\sum^{11}_{n = 1} (2 + 3^n )\]


Find the sum of the following series:

7 + 77 + 777 + ... to n terms;


Find the sum :

\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]


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).


A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.


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:

`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`


One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.


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 the following is also in G.P.:

a3, b3, c3


If a, b, c are in G.P., then prove that:

\[\frac{a^2 + ab + b^2}{bc + ca + ab} = \frac{b + a}{c + b}\]

If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.

  

Find the geometric means of the following pairs of number:

2 and 8


Find the geometric means of the following pairs of number:

a3b and ab3


If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.


Write the product of n geometric means between two numbers a and b

 


If abc are in G.P. and xy are AM's between ab and b,c respectively, then 


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.

1, –5, 25, –125 …


Check whether the following sequence is G.P. If so, write tn.

`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...


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?


For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r


Express the following recurring decimal as a rational number:

`2.3bar(5)`


Express the following recurring decimal as a rational number:

`51.0bar(2)`


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


Answer the following:

If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0


Answer the following:

If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2   


For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×