• Images
• Videos
• Documents
• Books
• Articles

Elements

The elements of an arithmetico-geometric sequence ( A n G n ) n ≥ 1 {\displaystyle (A_{n}G_{n})_{n\geq 1}} are the products of the elements of an arithmetic progression ( A n ) n ≥ 1 {\displaystyle (A_{n})_{n\geq 1}} (in blue) with initial value a {\displaystyle a} and common difference d {\displaystyle d} , A n = a + ( n − 1 ) d , {\displaystyle A_{n}=a+(n-1)d,} with the corresponding elements of a geometric progression ( G n ) n ≥ 1 {\displaystyle (G_{n})_{n\geq 1}} (in green) with initial value b {\displaystyle b} and common ratio r {\displaystyle r} , G n = b r n − 1 , {\displaystyle G_{n}=br^{n-1},} so that⁴

A 1 G 1 = a b A 2 G 2 = ( a + d ) b r A 3 G 3 = ( a + 2 d ) b r 2   ⋮ A n G n = ( a + ( n − 1 ) d ) b r n − 1 . {\displaystyle {\begin{aligned}A_{1}G_{1}&=\color {blue}a\color {green}b\\A_{2}G_{2}&=\color {blue}(a+d)\color {green}br\\A_{3}G_{3}&=\color {blue}(a+2d)\color {green}br^{2}\\&\ \,\vdots \\A_{n}G_{n}&=\color {blue}{\bigl (}a+(n-1)d{\bigr )}\color {green}br^{n-1}\color {black}.\end{aligned}}}

These four parameters are somewhat redundant and can be reduced to three: a b , {\displaystyle ab,} b d , {\displaystyle bd,} and r . {\displaystyle r.}

Example

The sequence

0 1 ,   1 2 ,   2 4 ,   3 8 ,   4 16 ,   5 32 , ⋯ {\displaystyle {\frac {\color {blue}{0}}{\color {green}{1}}},\ {\frac {\color {blue}{1}}{\color {green}{2}}},\ {\frac {\color {blue}{2}}{\color {green}{4}}},\ {\frac {\color {blue}{3}}{\color {green}{8}}},\ {\frac {\color {blue}{4}}{\color {green}{16}}},\ {\frac {\color {blue}{5}}{\color {green}{32}}},\cdots }

is the arithmetico-geometric sequence with parameters d = b = 1 {\displaystyle d=b=1} , a = 0 {\displaystyle a=0} , and r = 1 2 {\displaystyle r={\tfrac {1}{2}}} .

Series

Partial sums

The sum of the first n terms of an arithmetico-geometric series has the form

S n = ∑ k = 1 n A k G k = ∑ k = 1 n ( a + ( k − 1 ) d ) b r k − 1 = b ∑ k = 0 n − 1 ( a + k d ) r k = a b + ( a + d ) b r + ( a + 2 d ) b r 2 + ⋯ + ( a + ( n − 1 ) d ) b r n − 1 {\displaystyle {\begin{aligned}S_{n}&=\sum _{k=1}^{n}A_{k}G_{k}\\[5pt]&=\sum _{k=1}^{n}{\bigl (}a+(k-1)d{\bigr )}br^{k-1}\\[5pt]&=b\sum _{k=0}^{n-1}\left(a+kd\right)r^{k}\\[5pt]&=ab+(a+d)br+(a+2d)br^{2}+\cdots +{\bigl (}a+(n-1)d{\bigr )}br^{n-1}\end{aligned}}}

where A i {\textstyle A_{i}} and G i {\textstyle G_{i}} are the ith elements of the arithmetic and the geometric sequence, respectively.

This partial sum has the closed-form expression

S n = a b − ( a + n d ) b r n 1 − r + d b r ( 1 − r n ) ( 1 − r ) 2 = A 1 G 1 − A n + 1 G n + 1 1 − r + d r ( 1 − r ) 2 ( G 1 − G n + 1 ) . {\displaystyle {\begin{aligned}S_{n}&={\frac {ab-(a+nd)\,br^{n}}{1-r}}+{\frac {dbr\,(1-r^{n})}{(1-r)^{2}}}\\&={\frac {A_{1}G_{1}-A_{n+1}G_{n+1}}{1-r}}+{\frac {dr}{(1-r)^{2}}}\,(G_{1}-G_{n+1}).\end{aligned}}}

Derivation

Multiplying⁵

S n = a b + ( a + d ) b r + ( a + 2 d ) b r 2 + ⋯ + ( a + ( n − 1 ) d ) b r n − 1 {\displaystyle S_{n}=ab+(a+d)br+(a+2d)br^{2}+\cdots +{\bigl (}a+(n-1)d{\bigr )}br^{n-1}}

by r gives

r S n = a b r + ( a + d ) b r 2 + ( a + 2 d ) b r 3 + ⋯ + ( a + ( n − 1 ) d ) b r n . {\displaystyle rS_{n}=abr+(a+d)br^{2}+(a+2d)br^{3}+\cdots +{\bigl (}a+(n-1)d{\bigr )}br^{n}.}

Subtracting rSn from Sn, dividing both sides by b {\displaystyle b} , and using the technique of telescoping series (second equality) and the formula for the sum of a finite geometric series (fifth equality) gives

( 1 − r ) S n b = ( a + ( a + d ) r + ( a + 2 d ) r 2 + ⋯ + ( a + ( n − 1 ) d ) r n − 1 ) − ( a r + ( a + d ) r 2 + ( a + 2 d ) r 3 + ⋯ + ( a + ( n − 1 ) d ) r n ) = a + d ( r + r 2 + ⋯ + r n − 1 ) − ( a + ( n − 1 ) d ) r n = a + d ( r + r 2 + ⋯ + r n − 1 + r n ) − ( a + n d ) r n = a + d r ( 1 + r + r 2 + ⋯ + r n − 1 ) − ( a + n d ) r n = a + d r ( 1 − r n ) 1 − r − ( a + n d ) r n , S n = b 1 − r ( a − ( a + n d ) r n + d r ( 1 − r n ) 1 − r ) = a b − ( a + n d ) b r n 1 − r + d r ( b − b r n ) ( 1 − r ) 2 = A 1 G 1 − A n + 1 G n + 1 1 − r + d r ( G 1 − G n + 1 ) ( 1 − r ) 2 {\displaystyle {\begin{aligned}{\frac {(1-r)S_{n}}{b}}&=\left(a+(a+d)r+(a+2d)r^{2}+\cdots +{\bigl (}a+(n-1)d{\bigr )}r^{n-1}\right)-{\Bigl (}ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots +{\bigl (}a+(n-1)d{\bigr )}r^{n}{\Bigr )}\\[5pt]&=a+d\left(r+r^{2}+\cdots +r^{n-1}\right)-{\bigl (}a+(n-1)d{\bigr )}r^{n}\\[5pt]&=a+d\left(r+r^{2}+\cdots +r^{n-1}+r^{n}\right)-\left(a+nd\right)r^{n}\\[5pt]&=a+dr\left(1+r+r^{2}+\cdots +r^{n-1}\right)-\left(a+nd\right)r^{n}\\[5pt]&=a+{\frac {dr(1-r^{n})}{1-r}}-(a+nd)r^{n},\\[8pt]S_{n}&={\frac {b}{1-r}}\left(a-(a+nd)r^{n}+{\frac {dr(1-r^{n})}{1-r}}\right)\\[5pt]&={\frac {ab-(a+nd)br^{n}}{1-r}}+{\frac {dr(b-br^{n})}{(1-r)^{2}}}\\[5pt]&={\frac {A_{1}G_{1}-A_{n+1}G_{n+1}}{1-r}}+{\frac {dr(G_{1}-G_{n+1})}{(1-r)^{2}}}\end{aligned}}}

as claimed.

Infinite series

If −1 < r < 1, then the sum S of the arithmetico-geometric series, that is to say, the limit of the partial sums of the elements of the sequence, is given by⁶

S = ∑ k = 1 ∞ t k = lim n → ∞ S n = a b 1 − r + d b r ( 1 − r ) 2 = A 1 G 1 1 − r + d r G 1 ( 1 − r ) 2 . {\displaystyle {\begin{aligned}S&=\sum _{k=1}^{\infty }t_{k}=\lim _{n\to \infty }S_{n}\\[5pt]&={\frac {ab}{1-r}}+{\frac {dbr}{(1-r)^{2}}}\\[5pt]&={\frac {A_{1}G_{1}}{1-r}}+{\frac {drG_{1}}{(1-r)^{2}}}.\end{aligned}}}

If r is outside of the above range, b is not zero, and a and d are not both zero, the limit does not exist and the series is divergent.

Example

The sum

S = 0 1 + 1 2 + 2 4 + 3 8 + 4 16 + 5 32 + ⋯ {\displaystyle S={\frac {\color {blue}{0}}{\color {green}{1}}}+{\frac {\color {blue}{1}}{\color {green}{2}}}+{\frac {\color {blue}{2}}{\color {green}{4}}}+{\frac {\color {blue}{3}}{\color {green}{8}}}+{\frac {\color {blue}{4}}{\color {green}{16}}}+{\frac {\color {blue}{5}}{\color {green}{32}}}+\cdots } ,

is the sum of an arithmetico-geometric series defined by d = b = 1 {\displaystyle d=b=1} , a = 0 {\displaystyle a=0} , and r = 1 2 {\displaystyle r={\tfrac {1}{2}}} , and it converges to S = 2 {\displaystyle S=2} . This sequence corresponds to the expected number of coin tosses required to obtain "tails". The probability T k {\displaystyle T_{k}} of obtaining tails for the first time at the kth toss is as follows:

T 1 = 1 2 ,   T 2 = 1 4 , … , T k = 1 2 k {\displaystyle T_{1}={\frac {1}{2}},\ T_{2}={\frac {1}{4}},\dots ,T_{k}={\frac {1}{2^{k}}}} .

Therefore, the expected number of tosses to reach the first "tails" is given by

∑ k = 1 ∞ k T k = ∑ k = 1 ∞ k 2 k = 2. {\displaystyle \sum _{k=1}^{\infty }kT_{k}=\sum _{k=1}^{\infty }{\frac {\color {blue}k}{\color {green}2^{k}}}=2.}

Similarly, the sum

S = 0 ⋅ 1 6 5 6 + 1 ⋅ 1 6 1 + 2 ⋅ 1 6 6 5 + 3 ⋅ 1 6 ( 6 5 ) 2 + 4 ⋅ 1 6 ( 6 5 ) 3 + 5 ⋅ 1 6 ( 6 5 ) 4 + ⋯ {\displaystyle S={\frac {\color {blue}{0}\cdot \color {green}{\frac {1}{6}}}{\color {green}{\frac {5}{6}}}}+{\frac {\color {blue}{1}\cdot \color {green}{\frac {1}{6}}}{\color {green}{1}}}+{\frac {\color {blue}{2}\cdot \color {green}{\frac {1}{6}}}{\color {green}{\frac {6}{5}}}}+{\frac {\color {blue}{3}\cdot \color {green}{\frac {1}{6}}}{\color {green}{\left({\frac {6}{5}}\right)^{2}}}}+{\frac {\color {blue}{4}\cdot \color {green}{\frac {1}{6}}}{\color {green}{\left({\frac {6}{5}}\right)^{3}}}}+{\frac {\color {blue}{5}\cdot \color {green}{\frac {1}{6}}}{\color {green}{\left({\frac {6}{5}}\right)^{4}}}}+\cdots }

is the sum of an arithmetico-geometric series defined by d = 1 {\displaystyle d=1} , a = 0 {\displaystyle a=0} , b = 1 / 6 5 / 6 = 1 5 {\displaystyle b={\tfrac {1/6}{5/6}}={\tfrac {1}{5}}} , and r = 5 6 {\displaystyle r={\tfrac {5}{6}}} , and it converges to 6. This sequence corresponds to the expected number of six-sided dice rolls required to obtain a specific value on a die roll, for instance "5". In general, these series with d = 1 {\displaystyle d=1} , a = 0 {\displaystyle a=0} , b = p 1 − p {\displaystyle b={\tfrac {p}{1-p}}} , and r = 1 − p {\displaystyle r=1-p} give the expectations of "the number of trials until first success" in Bernoulli processes with "success probability" p {\displaystyle p} . The probabilities of each outcome follow a geometric distribution and provide the geometric sequence factors in the terms of the series, while the number of trials per outcome provides the arithmetic sequence factors in the terms.

Further reading

• D. Khattar. The Pearson Guide to Mathematics for the IIT-JEE, 2/e (New ed.). Pearson Education India. p. 10.8. ISBN 81-317-2876-5.
• P. Gupta. Comprehensive Mathematics XI. Laxmi Publications. p. 380. ISBN 81-7008-597-7.

References

1. "Arithmetic-Geometric Progression | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2021-04-21. https://brilliant.org/wiki/arithmetic-geometric-progression/ ↩

2. Swain, Stuart G. (2018). "Proof Without Words: Gabriel's Staircase". Mathematics Magazine. 67 (3): 209. doi:10.1080/0025570X.1994.11996214. ISSN 0025-570X. /wiki/Doi_(identifier) ↩

3. Edgar, Tom (2018). "Staircase Series". Mathematics Magazine. 91 (2): 92–95. doi:10.1080/0025570X.2017.1415584. ISSN 0025-570X. S2CID 218542483. /wiki/Doi_(identifier) ↩

4. K. F. Riley; M. P. Hobson; S. J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 978-0-521-86153-3. 978-0-521-86153-3 ↩

5. K. F. Riley; M. P. Hobson; S. J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 978-0-521-86153-3. 978-0-521-86153-3 ↩

6. K. F. Riley; M. P. Hobson; S. J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 978-0-521-86153-3. 978-0-521-86153-3 ↩