Arithmetic Progression

Definition:
A succession of numbers formed and arranged in a definite order according to certain definite rule is called progression.
If each term of a progression differs from the preceding term by a constant,then the progression is called arithmetic progression or AP.
The constant difference is called the common difference of the AP.
If first term is a and common difference is d then,
AP is a,(a+d), (a+2d), (a+3d),……

Formula:
Consider the AP a,(a+d), (a+2d), (a+3d),……
1.Tn = a+(n-1)d
2.Sn = (n/2)(2a+(n-1)d) or
Sn = (n/2)(first term + last term)

Where,
Tn is the nth term of the AP,
Sn is the sum of n terms in the AP,
a is the first term,
n is the number of term in the AP,
d is the common difference of the AP.

Example 1:
Find the 10 th term in the AP 2,6,10,14,……
Solution:
 Step 1:use the formula Tn = a+(n-1)d
 Step 2:In the given problem,
    a = first term = 2
    d= common difference = 4
  (to find d take any two terms in the AP and subtract it ( 6-2=4))
    n = 10
 Step 3:Apply the values in the formula.
   T10=2+(10-1)4
     =2+36
    =38
 Step 4:The 10 th term in the AP is 38

Example 2:
Find the sum of 5 terms in the AP 2,6,10,….
Solution:
 Step 1:use the formula sn = (n/2)(2a+(n-1)d)
 Step 2:In the given problem,
    a = first term = 2
    d= common difference = 4
  (to find d take any two terms in the AP and subtract it ( 6-2=4))
    n = 5
 Step 3:Apply the values in the formula.
    s10=(5/2)(2*2+(5-1)4)
    =2.5 * 20
    =50
 Step 4:The sum of 5 terms in the given AP is 50.