Summation Formulas

In mathematics, the summation is the basic addition of a sequence of numbers, called addends or summands; the result is their sum or total. The summation of an explicit sequence is denoted as a succession of additions.

For example, the summation of (1, 3, 4, 7) can be written as 1 + 3 + 4 + 7, and the result of this expression is 15, i.e., 1 + 3 + 4 + 7 = 15. Since the addition operation is both associative and commutative, parentheses are not necessary when listing the sequence, and the result will remain the same regardless of the order in which the summands are added.

Building on the concept of summation, a more compact and systematic way to represent a sum is through summation notation, also known as sigma (∑) notation. This notation allows us to express long sums concisely and can be applied to any formula or function.

The general form of summation notation is:

\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n

Where:

• ⅈ represents the first element in the sequence,
• n denotes the last element up to which the summation extends.

For example, \sum_{i=1}^{10} is a sigma notation representing the sum of the finite sequence 1 + 2 + 3 + 4 + ⋯ + 10, where the first element corresponds to i = 1 and the last element is n = 10.

Some common summation formulas are used in arithmetic series, geometric series, and more. Here are a few key types:

Where to Use Summation Formula?

Summation notation can be used in various fields of mathematics:

• Sequence and Series
• Integration
• Probability
• Permutation and Combination
• Statistics 

Note: A summation is a short form of repetitive addition. We can also replace summation with a loop of addition.

Properties of Summation

Property 1

_i=1∑ⁿ c = c + c + c + .... + c (n) times = nc 

For example: Find the value of  _i=1∑⁴ c.

By using property 1 we can directly calculate the value of _i=1∑⁴ c as 4×c = 4c.

Property 2

_c=1∑ⁿ kc = (k×1) + (k×2) +  (k×3)  + .... + (k×n) .... (n) times = k × (1 + ... + n)  = k  _c=1∑ⁿ c

For example: Find the value of _i=1∑⁴ 5i.

By using property 2 and 1 we can directly calculate value of _i=1∑⁴ 5i as 5 × _i=1∑⁴ i = 5 × ( 1 + 2 + 3 + 4) = 50.

Property 3

_c=1∑ⁿ (k+c) = (k+1) + (k+2) +  (k+3)  + .... + (k+n) .... (n) times = (n × k) +  (1 + ... + n)  = nk + _c=1∑ⁿ c

For example: Find the value of  _i=1∑⁴ (5+i).

By using property 2 and 3 we can directly calculate value of _i=1∑⁴ (5+i) as 5×4 +  _i=1∑⁴ i = 20 + ( 1 + 2 + 3 + 4) = 30.

Property 4

_k=1∑ⁿ (f(k) + g(k)) = _k=1∑ⁿ f(k) + _k=1∑ⁿ g(k)

For example: Find value of  _i=1∑⁴ (i + i²).

By using property 4  we can directly calculate value of _i=1∑⁴ (i + i²) as _i=1∑⁴ i  + _i=1∑⁴ i²  = ( 1 + 2 + 3 + 4) + (1 + 4 + 9 + 16) = 40.

Standard Summation Formulas

Here is a structured table summarizing all the given formula:

Related Reads:

• Sum of N Terms of AP Formula
• Sum of Natural Numbers
• Arithmetic Progression and Geometric Progression

Example on Summation Formula

Example 1: Find the sum of first 10 natural numbers, using the summation formula.
Solution: 

Using the summation formula for sum of n natural number _i=1∑ⁿ (i) = [n ×(n +1)]/2
We have sum of first 10 natural numbers =  _i=1∑¹⁰ (i) = [10 ×(10 +1)]/2 = 55

Example 2: Find the sum of 10 first natural numbers greater than 5, using the summation formula.
Solution: 

According to the question:

Sum of 10 first natural numbers greater than 5 =  _i=6∑¹⁵ (i) 

=  _i=1∑¹⁵ (i) -  _i=1∑⁵ (i) 
= [15 × 16 ] / 2 - [5 × 6]/2
= 120 - 15
= 105

Example 3: Find the sum of given finite sequence 1² + 2² + 3² + ... 8².
Solution: 

Given sequence is 1² + 2² + 3² + ... 8² , it can be written as  _i=1∑⁸ i² using the property/ formula of summation

_i=1∑⁸ i² = [8 ×(8 +1)× (2×8 +1)]/6 = [8 × 9 × 17] / 6 
= 204

Example 4: Simplify _c=1∑ⁿ kc.
Solution: 

Given summation formula = _c=1∑ⁿ kc
= (k×1) + (k×2) + ...... + (k×n) (n terms)
= k (1 + 2 + 3 +..... + n)
_c=1∑ⁿ kc = k _c=1∑ⁿ c

Example 5: Simplify and evaluate x₌₁∑ⁿ (4+x).
Solution: 

Given summation is  _x=1∑ⁿ (4+x)

As we know that _c=1∑ⁿ (k+c) = nk + _c=1∑ⁿ c
Given summation can be simplified as,
4n + _x=1∑ⁿ (x)

Example 6: Simplify _{x = 1}∑ⁿ (2x + x²).
Solution: 

Given summation is  _x=1∑ⁿ (2x+x²).

as we know that _k=1∑ⁿ (f(k) + g(k)) = _k=1∑ⁿ f(k) + _k=1∑ⁿ g(k)
given summation can be simplified as_x=1∑ⁿ (2x) + _x=1∑ⁿ (x²).