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definitions - Arithmetic_progression

arithmetic progression (n.)

1.(mathematics) a progression in which a constant is added to each term in order to obtain the next term"1-4-7-10-13- is the start of an arithmetic progression"

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synonyms - Arithmetic_progression

arithmetic progression (n.)

arithmetic series

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arithmetic progression (n.)


Wikipedia

Arithmetic progression

                   

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 of 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 (a_n) 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 sum of a finite arithmetic progression is called an arithmetic series.

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.

Contents

  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 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).[1]

So, for example, the sum of the terms of the arithmetic progression given by an = 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_1a_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 an = 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

  References

  1. ^ Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.95, ISBN 978-81-7434-480-9
  • Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-387-95419-8. 

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All translations of Arithmetic_progression


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