Geometric sequences

Geometric sequences
Prof.EvanBaros Profile Pic
Prof.EvanBaros,United Kingdom,Teacher
Published Date:26-07-2017
Your Website URL(Optional)
Comment
Geometric sequences MATH 1510 Lili Shen Geometric Definition Sequences A geometric sequence is a sequence of the form Mathematics of Finance 2 3 4 a; ar; ar ; ar ; ar ;:::: The nth term of a geometric sequence is given by n1 a = ar ; n where a is the first term and r is the common ratio of the sequence.Examples of geometric sequences MATH 1510 Lili Shen Geometric Example Sequences Mathematics (1) If a = 3 and r = 2, then we have the geometric of Finance sequence 2 3 4 3; 3 2; 3 2 ; 3 2 ; 3 2 ;:::; i.e., 3; 6; 12; 24; 48;:::: The nth term is n1 a = 3 2 : nExamples of geometric sequences MATH 1510 Lili Shen Geometric (2) The sequence Sequences Mathematics 2;10; 50;250; 1250;::: of Finance is a geometric sequence with a = 2 and r =5. When r is negative, the terms of the sequence alternate in sign. The nth term is n1 a = 2(5) : nExamples of geometric sequences MATH 1510 Lili Shen Geometric Sequences (3) The sequence Mathematics of Finance 1 1 1 1 1; ; ; ; ;::: 3 9 27 81 1 is a geometric sequence with a = 1 and r = . 3 The nth term is 1 a = : n n1 3Geometric sequences MATH 1510 Lili Shen Geometric If r 0 and r =6 1, then the points of the geometric Sequences sequence Mathematics of Finance 2 3 4 a; ar; ar ; ar ; ar ;::: lie in the graph of the exponential function x1 f(x) = ar : n1 If a 0, then the terms of the geometric sequence ar decrease when 0 r 1 and increase when r 1.Finding terms of a geometric sequence MATH 1510 Lili Shen Geometric Sequences Mathematics of Finance Example Find the common ratio, the first term, the nth term, and the eighth term of the geometric sequence 5; 15; 45; 135;::::Finding terms of a geometric sequence MATH 1510 Lili Shen Geometric Sequences Solution. Mathematics of Finance 45 The first term a = 5 and the common ratio r = = 3. 15 Then n1 a = 5 3 n and 7 a = 5 3 = 5 2187 = 10935: 8Partial sums of geometric sequences MATH 1510 Lili Shen For the geometric sequence Geometric Sequences 2 3 4 n1 a; ar; ar ; ar ; ar ;:::; ar ;:::; Mathematics of Finance the nth partial sum is n X k1 2 3 4 n1 S = ar = a+ ar + ar + ar + ar ++ ar : n k=1 If r = 1, then it is clear that S = na: nPartial sums of geometric sequences MATH 1510 Lili Shen To find a formula for S when r6= 1, note that n Geometric Sequences 2 3 n1 Mathematics S =a+ar+ar +ar +::: +ar ; n of Finance 2 3 n1 n rS = ar+ar +ar +::: +ar +ar : n Thus n S rS = a ar ; n n n (1 r)S = a(1 r ); n n a(1 r ) S = : n 1 rPartial sums of geometric sequences MATH 1510 Lili Shen Theorem Geometric Sequences n1 For the geometric sequence defined by a = ar , the nth n Mathematics of Finance partial sum n X k1 S = ar n k=1 is given by 8 na if r = 1; n S = n a(1 r ) a a 1 n+1 : = if r =6 1: 1 r 1 rFinding the partial sum of a geometric sequence MATH 1510 Lili Shen Geometric Sequences Mathematics of Finance Example Find the following partial sum of a geometric sequence: 1+ 4+ 16++ 4096:Finding the partial sum of a geometric sequence MATH 1510 Lili Shen Geometric Sequences Solution. Mathematics For this geometric sequence a = 1 and r = 4, thus of Finance n n1 a = 4 . 6 Since 4 = 4096, it follows that 7 1 4 1 16384 16383 S = = = = 5461: 7 1 4 1 4 3Infinite series MATH 1510 Lili Shen Geometric An expression of the form Sequences Mathematics 1 of Finance X a = a + a + a ++ a +::: n 1 2 3 n n=1 is called an infinite series. As n gets larger and larger, we are adding more and more of the terms of this series. Intuitively, as n gets larger, the partial sum S gets closer to the sum of the series. nInfinite series MATH 1510 Lili Shen 1 Consider the geometric sequence defined by a = . The n n Geometric 2 Sequences nth partial sum Mathematics of Finance 1 1 n+1 1 2 2 S = = 1 : n n 1 2 1 2 1 Since 0 as n1, S 1 as n1; that is, n n 2 lim S = 1: n n1Infinite series MATH 1510 Lili Shen 1 X Geometric In general, if the nth partial sum S of a series a gets n k Sequences k=1 Mathematics of Finance close to a finite number S as n gets large, i.e., lim S = S; n n1 we say that the infinite series converges (or is convergent). The number S is called the sum of the infinite series. If an infinite series does not converge, we say that the series diverges (or is divergent).Infinite geometric series MATH 1510 An infinite geometric series is a series of the form Lili Shen 1 X n1 2 3 4 n1 Geometric ar = a+ ar + ar + ar + ar ++ ar +:::: Sequences n=1 Mathematics of Finance Since the nth partial sum of such as series is n a(1 r ) S = n 1 r whenever r =6 1, it follows that a S as n1 n 1 r a wheneverjrj 1, and consequently is the sum of this r 1 infinite geometric series.Infinite geometric series MATH 1510 Lili Shen Theorem Geometric Sequences Ifjrj 1, then the infinite geometric series Mathematics of Finance 1 X n1 2 3 4 n1 ar = a+ ar + ar + ar + ar ++ ar +::: n=1 converges and has the sum a S = : 1 r Ifjrj 1, the series diverges.Infinite geometric series MATH 1510 Lili Shen Geometric Sequences Mathematics Example of Finance Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. 2 2 2 (1) 2+ + + +::: . 5 25 125 7 49 343 (2) 1+ + + +::: . 5 25 125Infinite geometric series MATH 1510 Lili Shen Solution. Geometric Sequences (1) This is an infinite geometric series with a = 2 and Mathematics 1 of Finance r = . Sincejrj 1, the series converges and the sum 5 is 2 5 S = = : 1 2 1 5 (2) This is an infinite geometric series with a = 1 and 7 r = . Sincejrj 1, the series diverges. 5Outline MATH 1510 Lili Shen Geometric Sequences Mathematics of Finance 1 Geometric Sequences 2 Mathematics of Finance

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.