Arithmetic Progression: Concepts & Tricks

Introduction
Arithmetic Progression or an Arithmetic Sequence is a sequence in which a constant is added to each term to get next term in the sequence. The constant term is called as the common difference of the A.P.
Suppose, a1,a2,a3,………an are the terms of an A.P.
Then,a2 = a1+d, a3 = a2+d , where a1 is the 1st term & d is the common difference which is constant for an Arithmetic Progression
In general,an+1 = an+d , where n is any natural number.
For example,
2, 5, 8, 11……… is an A.P with common difference 3.
3, 9, 15, 20…… is not an A.P because a4–a3 = 5 which is not equal to (a2–a1).
5, 1, -3, -7……..is an Arithmetic Progression with a common difference -4.
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Properties of Arithmetic Progressions
  1. If a constant is added to or is subtracted from each term of an Arithmetic Progression, then the resulting sequence is also an Arithmetic Progression with the same common difference.
  2. Each term of an Arithmetic Progression is multiplied by a constant or divided by a "non-zero" constant then resulting sequence is also an Arithmetic Progression.
a1,a2,a3……..an is an A.P,
It can be written as- a, a+ d, a + 2d, a + 3d … … a + (n – 1) d, where a is the first term, n is the number of terms and d is common difference.
Progression formula:
nth term of an A.P : an = a+(n–1)d
Sum of AP:  Sum of n terms of an A.P = n/2[2a + (n−1) d]
Arithmetic Progression formula:  n [2a + (n–1) d]/2 = n [a+ a + (n–1) d]/2 =n [a + an]/2
Illustration 1: 11th term of an A.P is 5 and 5th term of an A.P is 11, then what is 16th term?
Given, a11=5, a5 = 11
a + 10d =5—–(1)
a + 4d= 1—–(2)
Subtract (2) from (1) from gives,
6d =−6
d= −1
Putting d = -1 in ——(1), a – 10 = 5
a= 10+5= 15
a16 = a+15d
a16 = 15+5(−1) =0
Note: If mth term of an A.P is n and nth term of an Arithmetic Progression is m, then the common difference of the Arithmetic Progression is -1 and (m+n) th term is zero.
Arithmetic Mean
If number ‘c’ can be inserted between 2 numbers a and b such that a, c, b forms an Arithmetic Progression, then c is called the Arithmetic Mean of a and b
c–a = b–c
2c= a+b
c = (a+b)/2
Illustration 2: Which are the four numbers that can be inserted between 10 and 25 so that the resulting sequence forms an A.P?
Let 10,M1,M2,M3,M4, 25  be the resulting sequence.
Common difference, d = (25–10)/ (4+1) =15/5 = 3
M1 = 10+3= 13
M2 = 10+2*3 = 16
M3= 10+3*3 = 19
M4 = 10+4*3 =22
10, 13, 16, 19, 22, 25 is an Arithmetic Progression with common difference 3.
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