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Arithmetic Progression is a sequence of numbers in which the difference of every two consecutive numbers is always the same and constant. It is denoted as AP.
For example, take the natural numbers 1, 2, 3, 4, 5, 6,… is an AP, which has a common difference of 1. We can calculate it by taking the difference between two successive terms (say 1 and 2) equal to 1 (2 -1).
In AP, we have three main terms:
d=Common difference
a(n)=nth Term
S(n)=Sum of the first n terms
Common difference can be calculated as:
d = a(2) – a(1) = a(3) – a(2) = ……. = a(n) – a(n-1)
Where “d” is a common difference.
Terms can be assumed with respect to common difference:
a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d
where a= first term of the progeression and d=common difference.
nth Term of an AP:
a(n term) = a + (n − 1) × d
where a= first term,
n=number of terms,
d=common difference,
a(n term)= nth term.
Sum of n terms is given by:
S = n/2[2a + (n − 1) × d]
S= sum of nth term
a= first term,
n=number of terms,
d=common difference.
Example: Adding natural numbers up to 19 numbers.
AP = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19
Given, a = 1, d = 1 and an = 19
Now, by the formula we know;
S = n/2[2a + (n − 1) × d] = 15/2[2.1+(19-1).1]
S = 19/2[2+18] = 19/2 [20] = 19 x 10
S = 190
Hence, the sum of the first 19 natural numbers is 190.