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-   -   Some formulas of Arithmetic progression/series (http://mymathforum.com/algebra/45498-some-formulas-arithmetic-progression-series.html)

burgess July 28th, 2014 10:58 PM

Some formulas of Arithmetic progression/series
 
An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Example: 2,4,6,8,10…..

Arithmetic Series : The sum of the numbers in a finite arithmetic progression is called as Arithmetic series.
Example: 2+4+6+8+10…..

nth term in the finite arithmetic series

Suppose Arithmetic Series a1+a2+a3+…..an
Then nth term an=a1+(n-1)d

Where
a1- First number of the series
an- Nth Term of the series
n- Total number of terms in the series
d- Difference between two successive numbers

Sum of the total numbers of the arithmetic series

Sn=n/2*(2*a1+(n-1)*d)

Where
Sn – Sum of the total numbers of the series
a1- First number of the series
n- Total number of terms in the series
d- Difference between two successive numbers

Example:

Find n and sum of the numbers in the following series 3 + 6 + 9 + 12 + x?
Here a1=3, d=6-3=3, n=5

x= a1+(n-1)d = 3+(5-1)3 = 15

Sn=n/2*(2a1+(n-1)*d)
Sn=5/2*(2*3+(5-1)3)=5/2*18 = 45

I hope the above formulae are helpful to solve your math problems.

CRGreathouse July 29th, 2014 06:44 AM

In a related note, you may find the gp command sumformal() useful. Let's say you want to find the sum of $3n^2-2n$. Just type
Code:

sumformal(3*n^2-2*n)
and you get
Code:

n^3 + 1/2*n^2 - 1/2*n
in which you can substitute whatever values you need (either by hand or with gp's subst() command).

burgess August 6th, 2014 03:29 AM

Quote:

Originally Posted by CRGreathouse (Post 201718)
In a related note, you may find the gp command sumformal() useful. Let's say you want to find the sum of $3n^2-2n$. Just type
Code:

sumformal(3*n^2-2*n)
and you get
Code:

n^3 + 1/2*n^2 - 1/2*n
in which you can substitute whatever values you need (either by hand or with gp's subst() command).

Thanks for your suggestion :)

skipjack August 6th, 2014 09:58 AM

As the nth number is a1 + (n - 1)d, Sn = n(a1 + an)/2.

Note that (a1 + an)/2 is the median of the n numbers, and is the "middle" number of the series if n is odd.

It's slightly confusing to start by referring to a sequence of numbers and then keep switching between using the words "number" and "term".


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