Arithmetic progression

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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 5, 7, 9, 11, 13, … is an arithmetic progression with common difference of 2.

If the initial term of an arithmetic progression is <math>a_1</math> and the common difference of successive members is d, then the nth term of the sequence (<math>a_n</math>) is given by:

<math>\ a_n = a_1 + (n - 1)d,</math>

and in general

<math>\ a_n = a_m + (n - m)d.</math>

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

Template:Other uses-section The sum of the members of a finite arithmetic progression is called an arithmetic series.

Expressing the arithmetic series in two different ways:

<math> S_n=a_1+(a_1+d)+(a_1+2d)+\cdots+(a_1+(n-2)d)+(a_1+(n-1)d)</math>
<math> S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+\cdots+(a_n-2d)+(a_n-d)+a_n.</math>

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

<math>\ 2S_n=n(a_1+a_n).</math>

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

<math> S_n=\frac{n}{2}( a_1 + a_n).</math>

An alternate form results from re-inserting the substitution: <math>a_n = a_1 + (n-1)d</math>:

<math> S_n=\frac{n}{2}[ 2a_1 + (n-1)d].</math>

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).<ref>Aryabhatiya Template:Lang-mr, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.95, ISBN 978-81-7434-480-9</ref>

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

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

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

<math>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) },</math>

where <math>x^{\overline{n}}</math> denotes the rising factorial and <math>\Gamma</math> denotes the Gamma function. (Note however that the formula is not valid when <math>a_1/d</math> is a negative integer or zero.)

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

<math>m \times (m+1) \times (m+2) \times \cdots \times (n-2) \times (n-1) \times n \,\!</math>

for positive integers <math>m</math> and <math>n</math> is given by

<math>\frac{n!}{(m-1)!}.</math>

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

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

See also

References

<references/>

External links

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