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The summation is a process of adding up a sequence of given numbers, the result is their sum or total. It is usually required when large numbers of data are given and it instructs to total up all values in a given sequence. Summation is an important term in Mathematics as it calculates many terms of a given sequence. Summation notation is needed to represent large numbers. In other words, summation notation enables us to write short forms for the addition of very large numbers for a given date in a sequence.
A summation usually requires an infinite number of integrals. There can be two terms, thousands of terms, or many more. Few summations require infinite terms.
For these reasons, the summation is represented as [sum].
x 1+ x2 + x3 + x4 + x5 …… + xn = [sum_{i-n}^{n}]xi
Summation Representation
Generally, the Mathematical formulas need the addition of numerous variables. Summation or sigma notation is the easiest and simplest form of abbreviation used to give precise representation for a sum of the values of a variable.
Let y1, y2, y3, …yn represent a set of n numbers where y1 is the first number in the given set, and yi is the ith number in the given set.
Summation representation includes:
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The summation sign which is the Greek uppercase letter S is represented as a symbol [sum]. The summation symbol ([sum]) tells us to total up all the terms of a given sequence. A particular term of the sequence which is being summed is written at the right side of the summation symbol.
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The variable of summation, i.e. the variable which is being summed
The variable of summation is represented by an index which is set below the summation symbol. The index is usually denoted by i (other common variables used for the representation of the index are j and t.) The index resembles the expression as i = 1. The index usually undertakes values placed on the right – hand side of the equation and ends with the value above the summation sign.
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Some Important Summation Formulas
The general term of an arithmetic progression for the series of first n natural ie.e 1,2,3,4,5,…. is given as:
General term of an A.P. = a, a+ d, a + 2d, a + 3d…..
Arithmetic Progression Sum Formula
Arithmetic Progression sum formula for first n terms is given as
S = n/2 [ 2a + (n-1)d]
In the above arithmetic Progression sum formula:
n is the total number of terms, d is a common difference and a is the first term of the given series
The formula to calculate common difference ‘d’ in the arithmetic Progression sum formula is given as
Common Difference (d) = a2 – a1 = a3 –a2 an – an-1
Geometric Progression Sum Formula
For the given sequence :a1, a1r, a1r2,…….. a1rn-1, a1rn is given as
Geometric Progression sum formula (Sn) = a1 (rn -1)/ r-1 for r ≠ 1
Sum of Infinite Series Formula
Sum of an infinite series formula for the geometric formula with the common ratio r satisfying |r| < 1 is given as:
S∞ = [frac {a}{1-r}]
The notation for the above sum of geometric progression formula and sum of an infinite series formula is given as:
Sn = Sum of G.P with n terms
S∞= Sum of g.p with infinite terms
r = The common ratio
n = Total number of terms
a1 = The terms of the G.P sequance
The common ratio r is calculated as:
Common ratio (r) = [frac {a_2}{a_1} = frac {a_3}{a_2} = frac {a_n}{a_{n-1}}]
Summation of Cubes Formula
The summation of cubes formulas for first n natural number i.e. 13 + 23 + 33 + 43+ 53 ……….. + n3 is given as
{n(n+1)/2}²
Summation of n Numbers Formula
The sum of “n” numbers formulas for the natural numbers is given as
[frac {n(n+1)}{2}]
Sum of Even Numbers Formula
Sum of even numbers formulas for first n natural number is given
S = n(n + 1)
Sum of even numbers formula for first n consecutive natural numbers is given as
Se= n (n + 1)
Sum of Odd Numbers Formula
Sum of odd numbers formulas for first n natural number is given as
n²
Summation Representation Examples
[sum_{i=n}^{n}] yi = y 1+ y2 + y3……yn
[sum_{i=1}^{10}] yi = y3 + y4 + y5 + y6 + y7 + y8 + y9+ y10
[sum_{i=1}^{n}] x[_{i}^{2}] = x[_{1}^{2}] + x[_{2}^{2}] + x[_{3}^{2}] + ….. +x[_{n}^{2}]
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[sum] y – The limits of the summation generally appear as i = 1 through n. The representation below and above the summation symbol is usually omitted. Hence, this expression instructs us to total up the values of y, starting at y1 and ending with yn.
[sum] y = y 1+ y2 + y3……yn
[sum_{i=1}^{10}] yi = y 1+ y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9+ y10
Solved Examples
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Calculate the Value of [sum_{x-0}^{4}] y⁴
Solution:
[sum_{k=0}^{4}] = a₀ + [sum_{k=1}^{n}]
a₀ = 0⁴ = 0
= 0 + [sum_{n=1}^{4}] n⁴
= [sum_{n=1}^{4}] n⁴
Using the sum formula:
[sum_{k=1}^{n}] k⁴ = 1/30 n(n + 1)(2n + 1)(3n² + n+ 1)
= 1/30 × 4(4 + 1)(2 × 4 +1)(3 × 4² + 3 × 4 + 1)
= 354
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Find the Sum of the First 10 Odd Natural Numbers.
Solution:
Sequence – 1, 3, 5, 7, 9,11,……
The above given series is A.P., where
a= 1, d= 2, and n = 10
Sum of 10th term will be = n/2 [ 2a + (n-1)d]
S = 10/2[2×1 + (10 -1) × 2]
= 100
Hence, the sum of the first 10 odd natural numbers will be 100.
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Find the Value of [sum_{i=1}^{4}] with the Help of the Below Data
Solution
[sum_{i=1}^{4}] = x 1+ x2 + x3 + x4 = 1 + 2+ 3 + 4 = 10
Quiz Time
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The Series Which We Get by Adding the Terms of Geometric Sequence is Known as
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Harmonic series
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Geometric aries
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Arithmetic series
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Infinite Series
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For the Sequence 1,7, 25,79 ,241 ,727 the General Formula for an is
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3n+1 – 2
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3n– 2
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(-3) n + 3
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n2 – 2
Sum-Adding numbers
The term Sum in Math is used when two numbers are added together and the result is obtained. We also use this term in our daily lives. For example, when we go to the market and buy several things. We ask the shopkeeper to add the prices of all the items so that we can pay the money. The total amount of the prices of all items is called the sum. Here, you will learn more about the sum, how to find the sum in different situations.
Meaning of sum
In Maths, sum is the result obtained by adding two or more numbers. Thus, it is the property when we put different things together. In other words, sum is defined as when we put two or more numbers together to give a new result.
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Sum notation
When we add two or more numbers we use the sign of + (plus). The sum is the result obtained by adding two or more numbers. We can also represent the sum by using the symbol ? (sigma). This is used when we add a long list of numbers.
Sum of digits
Now, we will discuss how to find the sum of one-digit numbers, two-digit numbers, three-digit numbers and so on.
Sum of one digit, two-digit, and three-digit numbers
The sum of one-digit numbers can be found as 5 + 6= 11, the sum of two-digit numbers like 22+44=66, the sum of three digits like 456+124=580 and so on.