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- ...give the terms of a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </ma ...derived using the [[Geometric sequence#Infinite|sum formula for geometric series]] <cmath>\frac{1}{1-x} = \sum_{k=0}^{\infty} x^k = 1 + x + x^2 + x^3 + \do4 KB (659 words) - 11:54, 7 March 2022
- In [[algebra]], a '''geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between an ...er, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between consecutive terms varies.4 KB (649 words) - 20:09, 19 July 2024
- ...a_3 - a_2 = \cdots = a_n - a_{n-1}</math>. A similar definition holds for infinite arithmetic sequences. It appears most frequently in its three-term form: na ...arithmetic sequence. All infinite arithmetic series diverge. As for finite series, there are two primary formulas used to compute their value.4 KB (736 words) - 01:00, 7 March 2024
- 1</math> the [[series]] '''does not''' converge, but it can be extended to all expressed as an infinite product:9 KB (1,547 words) - 02:04, 13 January 2021
- ...o <math>1/10</math>. Using <math>a/(1-r)</math> for the sum of an infinite geometric sequence, we get <math>(15/100)/(1-(1/10)) = \boxed{\frac 16}</math>.3 KB (485 words) - 13:09, 21 May 2021
- ...rithmetic sequence of integers <math>a_1,a_2,\cdots</math> and an infinite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfying the following p ...the arithmetic sequence be <math>\{ a, a+d, a+2d, \dots \}</math> and the geometric sequence to be <math>\{ g, gr, gr^2, \dots \}</math>. Rewriting the problem5 KB (883 words) - 00:05, 2 June 2024
- ...times the sum of the original series. The common [[ratio]] of the original series is <math> \frac mn </math> where <math> m </math> and <math> n </math> are ...a new series, <math>a^2 + a^2 r^2 + a^2 r^4 + \ldots</math>. We know this series has sum <math>20050 = \frac{a^2}{1 - r^2}</math>. Dividing this equation3 KB (581 words) - 20:19, 22 September 2024
- ...10 times the sum of the original series. The common ratio of the original series is <math> \frac mn </math> where <math> m </math> and <math> n </math> are7 KB (1,119 words) - 20:12, 28 February 2020
- ...and <math>c</math> are [[positive]] [[integer]]s that form an increasing [[geometric sequence]] and <math>b - a</math> is the [[Perfect square|square]] of an in ...term of one of the series is <math>1/8</math>, and the second term of both series can be written in the form <math>\frac{\sqrt{m}-n}p</math>, where <math>m</7 KB (1,177 words) - 14:42, 11 August 2023
- .... To figure out which rational number, we sum an [[infinite]] [[geometric series]], <math>0.d25d25d25\ldots = \sum_{n = 1}^\infty \frac{d25}{1000^n} = \frac4 KB (584 words) - 13:38, 11 August 2024
- ...H</tt>'s there can be before the final five. This is an infinite geometric series whose sum is <math>\frac{3/64}{1-(15/32)}=\frac{3}{34}</math>, so the answe ...ere are <math>n</math> of that state in a row. We see that this gives us 2 geometric sequences: one with first term <math>\frac{1}{2^5}</math> and common ratio7 KB (1,087 words) - 12:09, 17 November 2024
- ...term of one of the series is <math>1/8</math>, and the second term of both series can be written in the form <math>\frac{\sqrt{m}-n}p</math>, where <math>m</ Let the second term of each series be <math>x</math>. Then, the common ratio is <math>\frac{1}{8x}</math>, and4 KB (710 words) - 15:06, 2 June 2024
- ...n notation'', provides a method for writing long, complicated, sometimes [[infinite]] sums neatly and compactly. Besides being easier to write than the explic For any constant c and [[finite]] (or [[absolutely convergent]]) [[series]] <math>a_k</math> and <math>b_k</math>,2 KB (335 words) - 16:17, 8 February 2024
- Using the formula for an infinite geometric series, we find912 bytes (145 words) - 09:51, 4 April 2012
- ...ath> and <math>g_n</math> are the <math>n</math>th terms of arithmetic and geometric sequences, respectively. The sum of the first <math>n</math> terms of an <math>\textbf{arithmetico-geometric sequence}</math> is <math>\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_2 KB (477 words) - 18:39, 17 August 2020
- ...aking dot products and using the formula for the sum of a finite geometric series, you can prove that the set of eigenvectors {<math>{f_n : n=0,\ldots, N-1}< ...s, we may for example take a function in <math>S</math> and write it as an infinite linear combination of the functions <math>F_n.</math> In finite Fourier an4 KB (724 words) - 18:15, 9 September 2006
- This is an infinite [[geometric series]], which sums to <math>\frac{\cos^0 \theta}{1 - \cos^2 \theta} = 5 \Longrig ...rficially less work. Again, applying the formula for an infinite geometric series,2 KB (231 words) - 23:18, 22 October 2024
- This is an infinite [[geometric series]] with common ratio <math>\frac{1}{x^3}</math> and initial term <math>x^{-1 ...hat<cmath>0.\overline{133}_n = \frac{n^2+3n+3}{n^3-1},</cmath>by geometric series.3 KB (480 words) - 13:50, 17 August 2020
- By adding the same constant to <math>20,50,100</math> a geometric progression results. The common ratio is: For the infinite series <math>1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\fra22 KB (3,348 words) - 11:53, 22 July 2024
- The geometric series <math>a+ar+ar^2\ldots</math> has a sum of <math>7</math>, and the terms inv The sum of an infinite geometric series is given by <math>\frac{a}{1-r}</math> where <math>a</math> is the first te2 KB (250 words) - 14:41, 27 July 2021