a.p.

Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.
An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), ...
where a = the first term , d = the common difference
Examples for Arithmetic Progressions
1, 3, 5, 7, ... is an arithmetic progression (AP) with a = 1 and d = 2
7, 13, 19, 25, ... is an arithmetic progression (AP) with a = 7 and d= 6


• n^th term of an arithmetic progression
t_n = a + (n – 1)d

where t_n = n^th term, a= the first term , d= common difference
Example 1 : Find 10^th term in the series 1, 3, 5, 7, ...
a = 1
d = 3 – 1 = 2
10^th term, t₁₀ = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19
Example 2 : Find 16^th term in the series 7, 13, 19, 25, ...
a = 7
d = 13 – 7 = 6
16^th term, t₁₆ = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97



• Number of terms of an arithmetic progression
n=(l−a)d+1

where n = number of terms, a= the first term , l = last term, d= common difference
Example : Find the number of terms in the series 8, 12, 16, . . .72
a = 8
l = 72
d = 12 – 8 = 4
n=(l−a)d+1=(72−8)4+1=644+1=16+1=17



• Sum of first n terms in an arithmetic progression
Sn=n2[ 2a+(n−1)d ] =n2[ a+l ]where a = the first term, d= common difference, l=tn=nth term = a+(n−1)d
Example 1 : Find 4 + 7 + 10 + 13 + 16 + . . . up to 20 terms
a = 4
d = 7 – 4 = 3
Sum of first 20 terms, S₂₀ =n2[ 2a+(n−1)d ]=202[ (2×4)+(20−1)3 ]
=10[8+(19×3)]=10[8+57]=650



Example 2 : Find 6 + 9 + 12 + . . . + 30
a = 6
l = 30
d = 9 – 6 = 3
n=(l−a)d+1=(30−6)3+1=243+1=8+1=9
Sum, S = n2[ a+l ]=92[ 6+30 ]=92×36=9×18=162


• Arithmetic Mean
If a, b, c are in AP, b is the Arithmetic Mean (A.M.) between a and c.

In this case, b=12(a+c)
 
The Arithmetic Mean (A.M.) between two numbers a and b = 12(a+b)
 
If a, a₁, a₂ ... a_n, b are in AP we can say that a₁, a₂ ... a_n are the n Arithmetic Means between a and b.
• If a, b, c are in AP, 2b = a + c
• To solve most of the problems related to A.P., the terms can be conveiently taken as
  
  3 terms : (a – d), a, (a +d)
  
  4 terms : (a – 3d), (a – d), (a + d), (a +3d)
  
  5 terms : (a – 2d), (a – d), a, (a + d), (a +2d)
• T_n = S_n - S_n-1
• If each term of an A.P. is increased, decreased , multiplied or divided by the same non-zero constant, the resulting sequence also will be in A.P.
• In an A.P., sum of terms equidistant from beginning and end will be constant