Summation Formulas

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 the large numbers of data are given and it instructs to total up all values in a given sequence. The 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 data 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 –aan - an-1


Geometric Progression Sum Formula

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= 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) =  a2 /a1 = a3 /a2 = an / an-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

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


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


1. 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


2. 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.


3. Find the Value of \[\sum_{i=1}^{4}\] with the Help of the Below Data

i

xi

1

1

2

2

3

3

4

4


Solution

\[\sum_{i=1}^{4}\] = x 1+ x2  + x3 + x4 = 1 + 2+ 3 + 4 = 10


Quiz Time


1. The Series Which We Get by Adding the Terms of Geometric Sequence is Known as

  1. Harmonic series

  2. Geometric aries

  3. Arithmetic series

  4. Infinite Series


2. For the Sequence 1,7, 25,79 ,241 ,727 the General Formula for an is

  1. 3n+1 - 2

  2. 3n- 2

  3. (-3) n + 3

  4. n2 - 2

FAQ (Frequently Asked Questions)

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 which 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) =  a2 /a1 = a3 /a2 = an / an-1

For example, the sequence 2,4,8,16,32… is a geometric sequence with common ratio 2

Common ratio (r) = 4/2 = 8/4 = 16/8 = 32/16 = 2.

The common ratio in geometric progression is represented as r.