Sum of Squares: Formula, Derivation & Examples

The sum of square numbers is known as the sum of squares. The sum of square denotes the square of two terms, three terms or n number of terms. The Sum of squares is a basic operation used in statistics, algebra and numbers series. In statistics, it is used to find the variation in the data. In Algebra and numbers series, it is used as a basic arithmetic operation.

Keyterms: Sum, Squares, Statistics, Algebra, Numbers series, Arithmetic operation, Numbers, Deviation

Read More: NCERT Solutions for Class 11 Math Sequences and Series

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Sum of Squares in Statistics

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In statistics, the sum of squares is also known as a total sum of squares and it is noted by TSS. The formula of a total sum of squares is given below,

TSS = ∑(xi - x‾)

Where,

Xi → Value in the set of numbers

x‾ → Mean of Numbers

i → 1,2,3,……, n

xi - x‾ → Deviation of Values

(xi + x‾) → Square of the Deviation

Also Read:

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Sum of Squares in Algebra

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Algebraic formulas are useful to calculate the squares of large numbers easily.

Let a and b be the two natural numbers then the Formula to find the sum of squares of a and b is

(a+b)² = a² + b² + 2ab

→ a² + b² = (a+b)² - 2ab

The below tables illustrates the proof of the above formula.

Sum of three squares in algebra

Let a, b and c be the three natural numbers, the formula to find the sum of squares of a, b and c is given as follows.

 (a+b+c)² = a²+b²+c²+2ab+2ac+2bc

→ a²+b²+c² = (a+b+c)²-2ab-2ac-2bc

The below tables illustrates the proof of the above formula.

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Sum of squares of natural numbers

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Natural numbers are numbers used for counting and ordering ranges from 1 to infinity and it is also known as positive integers. The formulas to calculate the sum of squares of n number of natural numbers, n even numbers and n odd numbers are tabulated below.

Read More: Sum of Squares Formula

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Formula Derivation of Sum of Squares

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The formula derivation of the sum of squares is given below.

Sum of squares of n natural numbers 

The formula is proved using the algebraic expression. The sum of the square of n natural numbers is denoted by ∑n². The algebraic expression used to prove this formula is 

a³ - b³ = (a-b) (a²+ ab + b²) 

Replace a as n and b as (n-1)

→ n³ - (n-1)³ = (n - (n-1))(n² + n(n-1) + (n-1)²)

→ n³ - (n-1)³ = 1(n² + n² - n + n² + 2n+1)

→ n³ - (n-1)³ = 3n² + n + 1

By substituting n= 1, 2…, n successively, we get

1³ – 0³ = 3 (1)² – 3 (1) + 1

2³ – 1³ = 3 (2)² – 3 (2) + 1

3³ – 2³ = 3 (3)² – 3 (3) + 1

Solving on both sides, we can get

→ n³ - 0³ = 3∑n² -3∑n + n

We know that, ∑n = (n(n+1))/2

→ ∑n² = (1/3) {n³ + (3n(n+1)/2) - n}

→ ∑n² = (1/6)(2n³ + 3n² + n)

Therefore,

∑n² = {n(n+1) (2n+1)}6

Sum of squares of n even numbers

The even number is denoted by 2n. The derivation of the sum of squares formula is derived below.

∑(2n)² = 2²+4²+6²+……..+(2n)²

→ ∑(2n)² = 2(1²+2²+3²+........+n²)

By applying a sum of squares of n natural number formula in the above equation We get,

→ ∑(2n)² = 2(( n x (n+1) x (2n+1))/6)

Therefore,

∑(2n)² ={2n (n+1) (2n+1)}3

Sum of squares of n odd numbers

The formula is derived by adding the sum of squares of 2n natural numbers and

n even numbers.

Σ(2n-1)² = ({(2n/6)(2n+1)}(4n+1))-((2n/3)(n+1)(2n+1))

→ Σ(2n-1)²=({(n/3)(2n+1)}(4n+1))-2(n+1)

→ Σ(2n-1)²=({(n/3)(2n+1)}(4n+1-2n-1)

Therefore,

Σ(2n-1)² = {n(2n+1) (2n-1)}3

Also Read:

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Things to Remember

• The sum of squares formulas is used to find the sum of squares of large numbers in an easy way. It is used in statistics to find the variance of a given value. In algebra and number series it is used as a basic arithmetic operation.
• The natural number is divided into two types, they are even numbers are odd numbers. The sum of two odd numbers is always an even number and the product of two or more odd numbers is always an odd number.
• Always we have to use algebraic expressions to derive the formula for the sum of squares.
• Sum of Squares of n natural numbers is given by {n(n+1) (2n+1)}6
• Sum of Squares of n even numbers is given by {2n(n+1) (2n+1)}3
• Sum of Squares of n odd numbers is given by {n(n+1) (2n-1)}3

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Previous Years’ Questions (PYQs)

1. If the 2nd, 5th and 9th terms of a non-constant… (JEE Main 2016)
2. If 4^x = 16^y = 64^z, then… (JKCET 2011)
3. Sum of n terms of the series 1 + 11…
4. Sum to 10 terms of the series 1+2…
5. The 10th common term between the series
6. The sum of squares of deviations for 10 observations taken from mean 50…
7. If S_m denotes the sum of first m terms of a G.P. of n terms
8. The 100th term of the sequence 1, 2, 2, 3, 3, 3,....
9. A sequence containing a finite number of terms is called a…
10. For a positive integer n let a(n)... (JEE Advanced 1999)
11. If the value of mode and mean is 60 …? (JEEA 2013)
12. The mean square deviation of a set of nn observation x₁,x₂,...x_n about a point cc is defined as 1n∑i=1n(xi−c)₂…? (BITS Admission Test 2015)
13. The arithmetic mean of the data 0,1,2,……,n with frequencies…?(BITS Admission Test 2015)
14. If coefficient of variation is 60 and standard deviation is 24, then Arithmetic mean is?  (KCET 2017)