Arithmatical progression

Quantity are said to be in arithmetical progression when they increase or decrease by a common difference . OR
the difference between two consecutive terms of a series is content is called the A.P. For example 3,7,11,15,.....................
8,2,-4,-10,...................
a,a+r,a+2r,a+3r,................
the common difference is found by subtracting any term of the series that which follows it .
# If you examine the series , a,a+r,a+2r,a+3r,............
We notice that any term that coefficient of d is always less by one then that number of terms in the series.
so, nth term of the series = a+(n-1)d;
where a is the first term and d is common difference of the series
# Sum of the number of terms of A.P.
let a is the first term,d the common difference , n the number of terms ,l the last term and s the required sum
s=a+(a+d)+(a+2d)+(a+3d)+..............+(l-2d)+(l-d)+l
s=l+(l-d)+l-2d)+(l-3d)+...............+(a+2d)+a+d)+a
by adding, 2s=(a+l)+(a+l)+(a+l)+.........to n terms
2s=n(a+l)
s=n/2(a+l) (i)
and you know that l=a+(n-1)d
so s = n/2[2a+(n-1)d] (ii)
#When three quantities are in A.P the middle term is said to be arithmetic mean of other two.
# Arithmetic mean:- let a and b be the two quantities and A is the arithmetic mean then, a,A,b are in A.P. We must have
A-a=b-A
2A=a+b
A=(a+b)/2
cont. for next post
in next some of theory and problem related to A.P.