What is Summation?
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:
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.
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.
The initiation point of the summation or the lower limit of the summation
The ending point of the summation or the upper limit of summation
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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 =This expression instructs us to total up all the value of y, starting at y1 and ending with yn.
\[\sum_{i=n}^{n}\] yi = y 1+ y2 + y3……yn
\[\sum_{i=3}^{10}\] yi = This expression instructs us to total up all the values of y, starting at y3 and ending with y10
\[\sum_{i=1}^{10}\] yi = y3 + y4 + y5 + y6 + y7 + y8 + y9+ y10
\[\sum_{i=1}^{n}\] x2i = This expression instructs us to total up squared values of x, starting at x1 and ending with xn.
\[\sum_{i=1}^{n}\] x\[_{i}^{2}\] = x\[_{1}^{2}\] + x\[_{2}^{2}\] + x\[_{3}^{2}\] + ….. +x\[_{n}^{2}\]
\[\sum\] 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 = This expression instructs us to total up the values of y, starting at y1 and ending with y10.
\[\sum_{i=1}^{10}\] yi = y 1+ y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9+ y10
Solved Examples
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
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.
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
The Series Which We Get by Adding the Terms of Geometric Sequence is Known as
Harmonic series
Geometric aries
Arithmetic series
Infinite Series
For the Sequence 1,7, 25,79 ,241 ,727 the General Formula for an is
3n+1 - 2
3n- 2
(-3) n + 3
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 𝚺 (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.
FAQs on Summation Formulas
1. What is the meaning of arithmetic progression?
Arithmetic Progression (AP) also known as the arithmetic sequence is a sequence that is different from each other by a common difference. We can calculate the common difference of any given arithmetic progression by calculating the difference between any two adjacent terms. Common Difference (d) = a2 - a1 = a3 – a2 an - an-1
For example, the sequence of 1,3,5,7,9, is an arithmetic sequence with the common difference of 2.
Common Difference (d) = 3 -1 = 2 , 5 - 3 = 2, 7 - 5 = 2
The common difference in the arithmetic progression is denoted as ‘d’.
2. What is the meaning of Geometric Progression?
A geometric progression (GP), also known as the geometric sequence is a sequence of numbers that varies from each other by a common ratio. We can calculate the common ratio of the given geometric sequence by finding the ratio between any two adjacent terms.
Common ratio (r) = \[\frac {a_2}{a_1} = \frac {a_3}{a_2} = \frac {a_n}{a_{n-1}}\]
For example, the sequence 2,4,8,16,32… is a geometric sequence with a common ratio of 2
Common ratio (r) = \[\frac {4}{2} = \frac {8}{4} = \frac {16}{8} = \frac {32}{16} =2\]
The common ratio in the geometric progression is represented as r.
3. Does sum mean add in Mathematics?
In Mathematics the term sum refers to the result obtained when we add numbers, objects, or things. For example, add 14 and6 add to make a sum of 20. Students can also follow Vedic Math ancient tricks of addition for summing up numbers. It is also useful in subtraction, division and multiplication.
4. At what age should children learn additional facts?
Children should be taught additional facts at the age of 6 itself. It can be started at home itself. The students get familiar with the facts which are useful for them in further classes. They can learn problems like word sums and multiple digit addition in further classes. It improves their counting skills and they aiso develop their mental ability at a very young age.
5. How can students score good marks in Math exams?
Some useful tips to score good marks in exam are:
Maintain separate register for formulae, theories and shortcut methods
Find solutions yourself
Understand the syllabus
Determine the areas of improvement
Keep the exam paper clean and tidy
Always give solutions in step
Attempt the questions which you are confident at
Draw pictures like graphs or geometrical shapes to make paper attractive.
6. How can students improve their Math skills?
Methods enabling the students to improve in Math are
Installing positivity and confidence
Scheduling practice
Using Tools that improve memory
Ask questions to test the level of understanding
Ensure firm basics
Focus on weaker topics.
7. Why do students have difficulty in Math?
The thing that makes Math difficult for many students is that it takes patience and persistence. for many students mats is not something that comes automatically.it takes plenty of effort to understand the multiple constructions.it requires a lot of time and energy. some children have butterflies in their stomachs when they think about Math as they seem to be prey challenging. Many children have diseases that affect Math learning skills and fear of failure results from some of them ignoring studying or they start crying before studying. Maths anxiety is a common problem among students which arises because of self-doubt and fear of failing.
8. How can parents improve Math skills among children at home itself?
Parents can introduce Math to their children at a very young age. Parents can create a positive environment at home so that children can learn math without fear and self-doubt. teach basic concepts of Math by fun or through playing techniques by solving Math puzzles or Math games. encourage children if they fail rather than scolding them which puts the morale of children down.
By using some below skills children Math can be improved:
Play with Math.
Practise Math in everyday life.
Mental Math training in the brain.
Brushing the basic concepts daily.
Online help or visual videos in practical form can help.