English

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

Advertisements
Advertisements

Question

For the following G.P.s, find Sn

0.7, 0.07, 0.007, .....

Sum
Advertisements

Solution

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
  Is there an error in this question or solution?
Chapter 2: Sequences and Series - Exercise 2.2 [Page 31]

APPEARS IN

RELATED QUESTIONS

Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.


The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.


Evaluate `sum_(k=1)^11 (2+3^k )`


If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.


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.


The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.


Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn


Find :

the 12th term of the G.P.

\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]


Find : 

nth term of the G.P.

\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]


Find the 4th term from the end of the G.P.

\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]


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


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


Evaluate the following:

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


Find the sum of the following series:

0.6 + 0.66 + 0.666 + .... to n terms


How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?


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 serie to infinity:

8 +  \[4\sqrt{2}\] + 4 + ... ∞


If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.


Find the rational number whose decimal expansion is `0.4bar23`.


Find the rational numbers having the following decimal expansion: 

\[3 . 5\overline 2\]


If a, b, c are in G.P., prove that log a, log b, log c are in A.P.


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:

(b + c) (b + d) = (c + a) (c + d)


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 a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.


If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.


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


If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then 


If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to 


In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is 


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

7, 14, 21, 28, …


Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1


Find the sum to n terms of the sequence.

0.2, 0.02, 0.002, ...


The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.


Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"` 


Answer the following:

Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...


Answer the following:

If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q


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


Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.


The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×