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Arithmetic Progression and Geometric Progression
Aman Bhati
January 25, 2015
Abstract
An article on arithmetic and geometric progression and proof of their summation formulas
Arithmetic Progression
1
Introduction
In mathematics, an arithmetic progression(AP) or arithmetic sequence is a sequence of
numbers such that the dierence between the consecutive terms is constant. For instance, the
sequence 5, 7, 9, 11, 13, 15 . . . is an arithmetic progression with common dierence of 2.
If the initial term of an arithmetic progression is a1 and the common dierence of successive
members is d, then the nth term of the sequence (an ) is given by:
an = a1 + (n − 1)d,
(1)
an = am + (n − m)d.
(2)
and in general
A nite portion of an arithmetic progression is called a nite arithmetic progression and
sometimes just called an arithmetic progression. The sum of a nite arithmetic progression is
called an arithmetic series.
The behavior of the arithmetic progression depends on the common dierence d. If the
common dierence is:
•
•
2
Positive, the members(terms) will grow towards positive innity.
Negative, the members(terms) will grow towards negative innity.
Summation
The sum of the members of a nite arithmetic progression is called an arithmetic series. For
example, consider the sum:
2 + 5 + 8 + 11 + 14 This sum can be found quickly by taking the number n of terms being
added (here 5), multiplying by the sum of the rst and last number in the progression (here
2 + 14 = 16), and dividing by 2:
n(a1 + an )
2
1
(3)
In the case above, this gives the equation:
2 + 5 + 8 + 11 + 14 =
an . For example:
− 23 + − 21 +
1
2
=
5(2+14)
2
3(− 32 + 12 )
2
=
5×16
2
= 40. This formula works for any real numbers a1 and
= − 32 .
2.1 Derivation
To derive the above formula, begin by expressing the arithmetic series in two dierent ways:
Sn = a1 + (a1 + d) + (a1 + 2d) + · · · + (a1 + (n − 2)d) + (a1 + (n − 1)d)Sn = (an − (n − 1)d) +
(an − (n − 2)d) + · · · + (an − 2d) + (an − d) + an . Adding both sides of the two equations, all
terms involving d cancel:
2Sn = n(a1 + an ). Dividing both sides by 2 produces a common form of the equation:
Sn = n2 (a1 + an ). An alternate form results from re-inserting the substitution: an =
a1 + (n − 1)d:
Sn =
n
[2a1 + (n − 1)d].
2
(4)
Furthermore the mean value of the series can be calculated via: Sn /n:
n
n = a1 +a
2 . In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya
(section 2.18). For more information on Aryabhata see [3]
3
Product
The product of the members of a nite arithmetic progression with an initial element a1 ,
common dierences d, and n elements in total is determined in a closed expression
a1 a2 · · · an = d
a n
a1 a1
a1
a1
Γ (a1 /d + n)
1
d( + 1)d( + 2) · · · d( + n − 1) = dn
= dn
d
d
d
d
d
Γ (a1 /d)
(5)
where xn denotes the rising factorial and Γ denotes the Gamma function. (Note however that
the formula is not valid when a1 /d is a negative integer or zero.)
This is a generalization from the fact that the product of the progression 1 × 2 × · · · × n is
given by the factorial n! and that the product
m × (m + 1) × (m + 2) × · · · × (n − 2) × (n − 1) × n
for positive integers m and n is given by
n!
(m−1)! .
Taking the example from above, the product of the terms of the arithmetic progression
given by an = 3 + (n − 1)(5) up to the 50th term is
P50 = 550 ·
Γ(3/5+50)
Γ(3/5)
≈ 3.78438 × 1098 .
2
4
Standard Deviation
The standard deviation of any arithmetic progression can be calculated via:
r
σ = |d|
(n − 1)(n + 1)
12
(6)
where n is the number of terms in the progression, and d is the common dierence between
terms.
5
Formulas at glance
If
a1 is the rst term of an arithmetic progression.
an is the nth term of an arithmetic progression.
d is the dierence between terms of the arithmetic progression.
l is the last term of an arithmetic progression.
n is the number of terms in the arithmetic progression.
Sn is the sum of n terms in the arithmetic progression.
n is the mean value of arithmetic series.
then
1. an = a1 + (n − 1)d (??)
2. an = am + (n − m)d (??)
3. Sn = n2 [2a1 + (n − 1)d] (4)
4. Sn = n(a21 +l) (3)
5. n = Sn /n
n
6. n = a1 +a
2
More information about Arithmetic Progression: [2]
3
Geometric Progression
1
Introduction
In mathematics, a geometric progression, also known as a geometric sequence, is a
sequence of numbers where each term after the rst is found by multiplying the previous one
by a xed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54,
... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric
sequence with common ratio 1/2.
Examples of a geometric sequence are powers rk of a xed number r, such as 2k and 3k .
The general form of a geometric sequence is
a, ar, ar2 , ar3 , ar4 , . . .
where r 6= 0 is the common ratio and a is a scale factor, equal to the sequence's start value.
2
Elementary Properties
The nth term of a geometric sequence with initial value a and common ratio r is given by
an = a rn−1
(7)
Such a geometric sequence also follows the recursive relation
an = r an−1 for every integer n ≥ 1. Generally, to check whether a given sequence is
geometric, one simply checks whether successive entries in the sequence all have the same
ratio.
The common ratio of a geometric sequence may be negative, resulting in an alternating
sequence, with numbers switching from positive to negative and back. For instance
1, -3, 9, -27, 81, -243, ...
is a geometric sequence with common ratio -3.
The behaviour of a geometric sequence depends on the value of the common ratio. If the
common ratio is:
• Positive, the terms will all be the same sign as the initial term.
• Negative, the terms will alternate between positive and negative.
• Greater than 1, there will be exponential growth towards positive or negative innity
(depending on the sign of the initial term).
• 1, the progression is a constant sequence.
• Between -1 and 1 but not zero, there will be exponential decay towards zero.
• -1, the progression is an alternating sequence
• Less than -1, for the absolute values there is exponential growth towards (unsigned)
innity, due to the alternating sign.
Geometric sequences (with common ratio not equal to -1, 1 or 0) show exponential growth
or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression
such as 4, 15, 26, 37, 48, . . . (with common dierence 11). This result was taken by T.R.
Malthus as the mathematical foundation of his Principle of Population. Note that the two
kinds of progression are related: exponentiating each term of an arithmetic progression yields
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a geometric progression, while taking the logarithm of each term in a geometric progression
with a positive common ratio yields an arithmetic progression.
An interesting result of the denition of a geometric progression is that for any value of the
common ratio, any three consecutive terms a, b and c will satisfy the following equation:
b2 = ac
(8)
where b is considered to be the geometric mean between a and c.
3
Summation
A geometric series is the sum of the numbers in a geometric progression. For example:
2 + 10 + 50 + 250 = 2 + 2 × 5 + 2 × 52 + 2 × 53 .
Letting a be the rst term (here 2), m be the number of terms (here 4), and r be the
constant that each term is multiplied by to get the next term (here 5), the sum is given by:
a(1 − rm )
1−r
In the example above, this gives:
2 + 10 + 50 + 250 =
2(1−54 )
1−5
=
−1248
−4
(9)
= 312.
The formula works for any real numbers a and r (except r = 1, which results in a division
by zero). For example:
3)
3)
−2π + 4π 2 − 8π 3 = −2π + (−2π)2 + (−2π)3 = −2π(1−(−2π)
= −2π(1+8π
≈ −214.855.
1+2π
1−(−2π)
3.1 Derivation
To derive
this formula, rst write a general geometric series as:
P
n
k−1
k=1 ar
= ar0 + ar1 + ar2 + ar3 + · · · + arn−1 .
We can nd a simpler formula for this sum by multiplying both sides of the above equation
by (1-r) and we'll see that
(1 − r)
n
X
ark−1 = (1 − r)(ar0 + ar1 + ar2 + ar3 + · · · + arn−1 )
k=1
= ar0 + ar1 + ar2 + ar3 + · · · + arn−1 − ar1 − ar2 − ar3 − · · · − arn−1 − arn
= a − arn
(10)
since all the other terms cancel. If r 6= 1, we can rearrange the above to get the convenient
formula for a geometric series that computes the sum of n terms:
Pn
k=1 ar
k−1
=
a(1−rn )
1−r .
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3.2 Related Formulas
If one were to begin the sum not from k=0, but from a dierent value, say m, then
Pn
a(rm −rn+1 )
k
.
k=m ar =
1−r
Dierentiating this formula with respect to r allows us to arrive at formulae for sums of the
formP
n
s k
k=0 k r .
For example:
n
Pn
d Pn
k
k−1 = 1−rn+1 − (n+1)r .
k=0 r =
k=1 kr
dr
1−r
(1−r)2
For a geometric
series containing only even powers of r, multiply by 1 − r2 :
P
n
(1 − r2 ) k=0 ar2k = a − ar2n+2 .
Then
n
X
a(1 − r2n+2 )
.
1 − r2
ar2k =
k=0
(11)
Equivalently, take r2 as the common ratio and use the standard formulation.
For a series P
with only odd powers of r
n
2k+1
k=0 ar
(1 − r2 )
and
= ar − ar2n+3
n
X
ar2k+1 =
k=0
ar(1 − r2n+2 )
.
1 − r2
(12)
3.3 Innite Geometric Series
An innite geometric series is an innite series whose successive terms have a common
ratio. Such a series converges if and only if the absolute value of the common ratio is less than
one (|r| < 1). Its value can then be computed from the nite sum formulae
P∞
k=0 ar
k
= limn→∞
Pn
k=0 ar
k
= limn→∞
a(1−rn+1 )
1−r
=
a
1−r
− limn→∞
arn+1
1−r
Since:
rn+1 → 0 as n → ∞ when |r| < 1.
Then:
∞
X
k=0
ark =
a
a
−0=
1−r
1−r
(13)
For P
a series containing only even powers of r,
∞
2k
k=0 ar
=
a
1−r2
and for odd powers only,
P∞
k=0 ar
2k+1
=
ar
1−r2
In cases where the sum does not start at k = 0,
P∞
k=m ar
k
=
arm
1−r
The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach
algebra, as long as the norm of r is less than one, and also in the eld of p-adic numbers if
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|r|p < 1. As in the case for a nite sum, we can dierentiate to calculate formulae for related
sums. For example,
P∞
P∞
1
= (1−r)
2
This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,
P∞ 2 k
P∞
r(1+4r+r2 )
r(1+r) P∞
r
3 k
k
k=0 k r = (1−r)3 ;
k=0 k r =
k=0 kr = (1−r)2 ;
(1−r)4
d
dr
k=0 r
k
=
k=0 kr
k−1
Also, the innite series 1/2 + 1/4 + 1/8 + 1/16 + ... is an elementary example of a series
that converges absolutely.
It is a geometric series whose rst term is 1/2 and whose common ratio is 1/2, so its sum is
1/2
1
1
1
1
2 + 4 + 8 + 16 + · · · = 1−(+1/2) = 1.
The inverse of the above series is 1/2 - 1/4 + 1/8 - 1/16 + ... is a simple example of an
alternating series that converges absolutely. It is a geometric series whose rst term is 1/2 and
whose common ratio is - 1/2, so its sum is
1/2
1
1
1
1
1
2 − 4 + 8 − 16 + · · · = 1−(−1/2) = 3 .
3.4 Complex Numbers
The summation formula for geometric series remains valid even when the common ratio is a
complex number. In this case the condition that the absolute value of r be less than 1 becomes
that the modulus of r be less than 1. It is possible to calculate the sums of some non-obvious
geometric series. For example, consider the proposition
P∞
k=0
sin(kx)
rk
=
r sin(x)
1+r2 −2r cos(x)
The proof of this comes from the fact that
sin(kx) =
eikx − e−ikx
,
2i
(14)
which is a consequence of Euler's formula. Substituting this into the original series gives
∞
X
sin(kx)
k=0
rk
#
" ∞ k
∞ −ix k
X
e
1 X eix
−
.
=
2i
r
r
k=0
(15)
k=0
This is the dierence of two geometric series, and so it is a straightforward application of the
formula for innite geometric series that completes the proof.
4
Product
The product of a geometric progression is the product of all terms. If all terms are positive,
then it can be quickly computed by taking the geometric mean of the progression's rst and
last term, and raising that mean to the power given by the number of terms. (This is very
similar to the formula for the sum of terms of an arithmetic sequence: take the arithmetic
mean of the rst and last term and multiply with the number of terms.)
n
Y
√
ari = ( a1 · an+1 )n+1 (if a, r > 0).
i=0
7
(16)
Proof:
Let the product be represented by P:
P = a · ar · ar2 · · · arn−1 · arn .
Now, carrying out the multiplications, we conclude that
P = an+1 r1+2+3+···+(n−1)+(n) .
Applying the sum of arithmetic series, the expression will yield
n(n+1)
2
P = an+1 r
n
P = (ar 2 )n+1 .
.
We raise both sides to the second power:
P 2 = (a2 rn )n+1 = (a · arn )n+1 .
Consequently
P 2 = (a1 · an+1 )n+1 and
n+1
P = (a1 · an+1 ) 2 , which concludes the proof.
More information on Geometric Progression : [1]
References
[1] Thomas L. Heath. The Thirteen Books of Euclid's Elements. Cambridge University Press,
second edition, 1956.
[2] Laurence E. Sigler. Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo
Pisano's Book of Calculation. Springer-Verlag, second edition, 2002.
[3] Wikipedia. Aryabhata:wiki. http://en.wikipedia.org/wiki/Aryabhata, 2015.
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