Summation Formula with Solved Example

Summation is required when a large number of data are concerned. In other words, in order to write a very large number, summation notation is useful. We frequently need to calculate a large number of terms in a sequence. As a result, methods for summarizing a series are crucial in mathematics. Summation notation is highly beneficial for writing a really large number. In simple words, summation notation helps write a short form for the addition of a very large number of data. We use this symbol –, which is called sigma in order to denote summation. Here, we will be discussing the summation formula along with some solved examples and important questions. 

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What is Summation?

A summation, or sum, is the result of adding all the numbers or quantities supplied in the form of a series arithmetically. An integral number of terms is always present in a summation. It is possible to have as few as two terms or as many as a thousand or even more. Some summations have an unlimited number of terms.

As a result, the summation symbol was created, i.e.

x₁+x₂+x₃+x₄+x₅+……... x_n = ∑ⁿ_i-n  x_i

In this section, we'll go over summation notation, often known as sigma notation. We'll start with two integers, n, and m, along with n < m and a list of numbers denoted as follows:

a_n, a_n+1, a_n+2, …, a_m-2, a_m-1, a_m

a_n+a_n+1+a_n+2+…+a_m-2+a_m-1+a_m

This is a complicated notation for huge lists. The aforesaid scenario is depicted as follows.

∑^m_i=n a_i =a_n+ a_n+1, a_n+2 +…+ a_m-2, a_m-1, a_m

i is known as the summation index. This syntax instructs us to add all of the a_i's values.

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Some Formulas Related to summation

The total of the series is calculated using summation formulas. There are many distinct sorts of sequences, such as arithmetic, geometric, and so on, and thus many different types of summing formulas for diverse sequences. There are also summation formulas for finding the sum of natural numbers, the sum of natural numbers squared, the sum of natural numbers cubes, the sum of even numbers, the sum of odd numbers, and so on.

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Summarization is frequently required when dealing with vast amounts of data. Summation notation is useful for writing a huge number. Summation is the sequence [1,2,4, 2...] whose value is the sum of each integer in the sequence. To put it another way, summation notation aids in the creation of a short form for the addition of a big quantity of data.

To express summation, we use the sign –, often known as sigma. When a sequence is needed to add from left to right, it could provide a partial sum, running total, or prefix sum as an intermediate result.

For the series of first n natural numbers, i.e., 1,2,3,4,5, ……. the standard word for an arithmetic progression is:

A.P. general word = a, a+ d, a + 2d, a + 3d.....

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Sum Formula for Arithmetic Progression

For the first n terms, the arithmetic progression sum formula is as follows:

S = n/2 2a+(n - 1) d

Progression sum formula in the above arithmetic:

The total number of terms is n, the common difference is d, and the first term in the series is a.

In the arithmetic Progression sum formula, the formula for calculating common difference 'd' is as follows:

Common Difference (d) = a₂ – a₁ = a₃ –a₂ a_n –a_n-1

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Sum of Geometric Progression Formula

For the above sequence, the sum formula for geometric progression is:

a₁, a₁ r, a₁ r², ........a₁ r_n-1, a₁ rⁿ is written as:

Sum formula for geometric progression (S_n) = a₁ (rⁿ-1)/ r-1 for r≠1

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Sum of Infinite Series Formula

For the geometric formula with the common ratio r satisfying |r| < 1, the sum of an infinite series formula is:

S∞ = a/1 – r

The sum of the geometric progression formula and the sum of an infinite series formula is written as follows:

Sn is the sum of the G.P terms with n terms.

S∞= Sum of g.p with infinite terms

The common ratio is denoted by the letter r.

n denotes the total number of terms.

a₁ = The G.P sequence's terms

r is the common ratio, which is determined as follows:

Common ratio (r) = a₂ /a₁ = a₃ /a₂ = a_n / a_n-1

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Summation of Cubes Formula

For the first n natural numbers, 13 + 23 + 33 + 43+ 53........... + n3, the summing of cubes formulas are as follows:

{n(n+1)/2}²

Summation of n Numbers Formula

For natural numbers, the sum of "n" numbers formulas are as follows:

n (n + 1)/2

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Sum of Even Numbers Formula

The formulas for the sum of even numbers for the first n natural numbers are presented.

S = n (n + 1)

For the first n consecutive natural numbers, the sum of even numbers formula is as follows:

Se = n (n + 1)

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Sum of Odd Numbers Formula

For the first n natural numbers, the sum of odd numbers formulas are as follows:

n²

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Things to Remember

• Summation or sigma notation is considered to be the easiest and simplest form of an abbreviation that is used to give precise representation for a sum of the values of a variable.
• The summation sign, which is the Greek uppercase letter S, is denoted by a symbol ∑. The summation symbol (∑) suggests totaling up all the terms of a given sequence. 
• General term of an A.P. = a, a+ d, a + 2d, a + 3d…..
• Sum formula for geometric progression (S_n) = a₁ (rⁿ-1)/ r-1 for r≠1
• For the first n natural numbers, 13 + 23 + 33 + 43+ 53........... + n3, the summing of cubes formulas are as follows: {n(n+1)/2}²?

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