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Summation formula
Summation formulas are used to calculate the summation of a sequence. Here, the summation is the sum of consecutive series of a sequence.
as mentioned earlier here, summation refers to the addition of consecutive series of a sequence. For example, let’s assume there are n terms in a sequence, and all the terms are expressed as, a1, a2 up to an. There are different sequences and series for which several formulas are used. These sequences and series include arithmetic progression, geometric progression, harmonic progression etc.
The formulas of summation are
The formula of summation for the first n natural number is,
∑ 1 + 2 + 3 + … + n = n * (n+1) / 2
The summation of the square of first n natural numbers is,
∑ 12 + 22 + 32 + … + n2 = n * (n+1) * (2n+1) / 6
The summation of the cube of the first n natural numbers is,
∑13 + 23 + 33 + … + n3 = n2 (n+1)2 / 4
The summation of fourth power of first n natural numbers is,
∑ 14 + 24 + 34 + … + n4 = 1/30 * n* (n + 1) * (2n + 1) * (3n2 + 3n − 1)
The summation of first n even natural numbers is,
∑ 2 + 4 + 6 + … nth term = n (n + 1)
The summation of first n odd natural numbers is,
∑ 1 + 3 + 5 + …. (Nth term) = n2
The summation of n terms in an arithmetic progression (in this sequence the numbers are such as a, a + d, a + 2d, a + 3d … a + (n – 1) * d etc) is,
∑ a + (i−1) * d = n / 2 * [2a + (n − 1) * d]
In geometric progression (in this series, the numbers are such as a, a * r, a * r2, a*r(n-1)
The summation of the first n term in the geometric progression is,
∑ a * ri-1 = a * (1−rn) / (1−r)
The summation of infinite terms in a geometric progression is,
∑ a * (ri-1) = a / (1−r) (only when |r| < 1)
Solved example
What is the summation of all even numbers from 1 to 1000?
Answer. The number of even number from 1 to 100 is 50
Using the formula, the summation is,
n * (n + 1)
= 50 * (50 + 1)
= 2550
2. Find the summation of first 10 natural numbers.
Answer. The summation is, 10 * (10 + 1) / 2
= 55
