User talk:Syaffuan

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is $a_{1}$ and the common difference of successive members is d, then the nth term of the sequence is given by:

$\ a_{n}=a_{1}+(n-1)d,$

and in general

$\ a_{n}=a_{m}+(n-m)d.$

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

• Positive, the members (terms) will grow towards positive infinity.
• Negative, the members (terms) will grow towards negative infinity.

Sum

The sum of the members of a finite arithmetic progression is called an arithmetic series.

Expressing the arithmetic series in two different ways:

$S_{n}=a_{1}+(a_{1}+d)+(a_{1}+2d)+\cdots +(a_{1}+(n-2)d)+(a_{1}+(n-1)d)$

$S_{n}=(a_{n}-(n-1)d)+(a_{n}-(n-2)d)+\cdots +(a_{n}-2d)+(a_{n}-d)+a_{n}.$

Adding both sides of the two equations, all terms involving d cancel:

$\ 2S_{n}=n(a_{1}+a_{n}).$

Dividing both sides by 2 produces a common form of the equation:

$S_{n}={\frac {n}{2}}(a_{1}+a_{n}).$

An alternate form results from re-inserting the substitution: $a_{n}=a_{1}+(n-1)d$:

$S_{n}={\frac {n}{2}}[2a_{1}+(n-1)d].$

In 499 CE Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18) .

So, for example, the sum of the terms of the arithmetic progression given by a_n = 3 + (n-1)(5) up to the 50th term is

$S_{50}={\frac {50}{2}}[2(3)+(49)(5)]=6,275.$

Product

The product of the members of a finite arithmetic progression with an initial element a₁, common differences d, and n elements in total is determined in a closed expression

$a_{1}a_{2}\cdots a_{n}=d^{n}{\left({\frac {a_{1}}{d}}\right)}^{\overline {n}}=d^{n}{\frac {\Gamma \left(a_{1}/d+n\right)}{\Gamma \left(a_{1}/d\right)}},$

where $x^{\overline {n}}$ denotes the rising factorial and $\Gamma$ denotes the Gamma function. (Note however that the formula is not valid when $a_{1}/d$ is a negative integer or zero.)

This is a generalization from the fact that the product of the progression $1\times 2\times \cdots \times n$ is given by the factorial $n!$ and that the product

$m\times (m+1)\times (m+2)\times \cdots \times (n-2)\times (n-1)\times n\,\!$

for positive integers $m$ and $n$ is given by

${\frac {n!}{(m-1)!}}.$

Taking the example from above, the product of the terms of the arithmetic progression given by a_n = 3 + (n-1)(5) up to the 50th term is

$P_{50}=5^{50}\cdot {\frac {\Gamma \left(3/5+50\right)}{\Gamma \left(3/5\right)}}\approx 3.78438\times 10^{98}.$

See Also

• Geometric progression
• Generalized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowing several possible differences.
• Harmonic progression

References

• Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-387-95419-8.

External links

• Weisstein, Eric W. "Arithmetic progression". MathWorld.
• Weisstein, Eric W. "Arithmetic series". MathWorld.