Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 years ago

# Prove the relation: 1² + 2² + 3² ... n² = 1/6 n(n+1)(2n+1)?

I know that the above statement can be proved by Mathematical Induction. Proving this way only proves correctness of the formula. I was wondering how people found out that the sum of perfect squares is equal to n(n+1)(2n+1)/6.

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• 9 years ago

One can prove sum of all powers of natural numbers one by one first taking power 1, i.e., 1 + 2 + 3 + ..., then power 2, i.e., 1^2 + 2^2 + 3^2 + ... and so on by a general method described as under.

n^3 - (n - 1)^3 = n^3 - (n^3 - 3n^2 + 3n - 1)

=> n^3 - (n - 1)^3 = 3n^2 - 3n + 1

Plugging n = 1, 2, 3, ... n

1^3 - 0^3 = 3 * 1^2 - 3 *1 + 1

2^3 - 1^3 = 3 * 2^2 - 3 * 2 + 1

3^3 - 2^3 = 3 * 3^2 - 3 * 3 + 1

.....

,.....

n^3 - (n - 1)^3 = 3 * n&2 - 3 * n + 1

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n^3 = 3 * (1^2 + 2^2 + 3^2 + ... + n^2) - 3 * (1 + 2 + 3 + ... + n) + n

=> 3 * (1^2 + 2^2 + 3^2 + ... + n^2)

= n^3 - 3 * (1 + 2 + 3 + .... + n) - n

= n^3 + (3/2) n (n + 1) - n

= (1/2) (2n^3 + 3n^2 + 3n - 2n)

= (1/2) (2n^3 + 3n^2 + n)

= (n/2) (2n^2 + 3n + 1)

= (n/2) (n + 1) (2n + 1)

=> (1^2 + 2^2 + 3^2 + ... n^2)

= (1/6) n (n + 1) (2n + 1).

The sum 1^3 + 23 + 3^3 + ... + n^3

can similarly be proved taking

n^4 - (n - 1)^4 = 4n^3 - 6n^2 + 4n - 1

Thus, one after another, formula for sum of any power of the natural numbers can be derived.

Refer to my answer as under where the sum 1^4 + 2^4 + 3^4 + ... + n^4 is derived.

• 9 years ago

One possible way is to consider the relation (r+1)^3 - r^3 =3r^2+3r+1

If you sum each side from r=1 to r=n you get (n+1^3 - 1=3S+3n(n+1)/2+n , where S is the required sum

and you get 3S=(n+1)^3-1-3n(n+1)/2-n=n^3+(3/2)n(n+1)-n=(n/2)(2n^2+3n+1)=(1/2)n(n+1)(2n+1`)

so S=(1/6)n(n+1)(2n+1)

This depends on knowing the sum of r from 1 to n but this is an arithmetic series and is well known.

You also should be aware that the sum of [u(r+1)-u(r)] from r=1 to r=n is u(n+1)-u(n)

You can then use the same idea to obtain the sum of r^3 using ((r+1)^4 -r^4)) etc.

• 9 years ago

Form a recurrence relation and find the formula by comparing coefficients:

a(n + 1) = a(n) + (n + 1)²

Guess a(n) = an³ + bn² + cn + d

a(n + 1)³ + b(n + 1)² + c(n + 1) + d = an³ + bn² + cn + d + (n + 1)²

a(n³ + 3n² + 3n + 1) + b(n² + 2n + 1) + c(n + 1) + d = an³ + bn² + cn + d + n² + 2n + 1

an³ + 3an² + 3an + a + bn² + 2bn + b + cn + c + d = an³ + bn² + cn + d + n² + 2n + 1

3an² + (3a + 2b)n + (a + b + c) = n² + 2n + 1

3a = 1

a = ⅓

3a + 2b = 2

2b = 2 - 3a

b = 1 - 3a / 2

b = 1 - ½

b = ½

a + b + c = 1

c = 1 - a - b

c = 1 - ⅓ - ½

c = 1 / 6

Guess a(n) = n³ / 3 + n² / 2 + n / 6 + d

When n = 0, a(n) = 0

d = 0

a(n) = n³ / 3 + n² / 2 + n / 6

a(n) = (2n³ + 3n² + n) / 6

a(n) = n(2n² + 3n + 1) / 6

a(n) = n(n + 1)(2n + 1) / 6

What is with the thumbs down, I used a perfectly valid method.

• 9 years ago

2³=(1+1)³=1³+3*1²+3*1+1

3³=(2+1)³=2³+3*2²+3*2+1

4³=(3+1)³=3³+3*3²+3*3+1

5³=(4+1)³=4³+3*4²+3*4+1

...

n³=((n-1)+1)³=(n-1)³+3(n-1)²+3(n-1)+1

-----------------------------

SUM

2³+3³+4³+5³+...+n³ =[1³+2³+3³+4³+5³+...+(n-1)³]+ 3[1²+2²+3²+4²+5²+...+(n-1)²]+ 3[1+2+3+4+5+...+(n-1)]+ [1+1+1+1+1+...+1]

n³ =[1³]+ 3[1²+2²+3²+4²+5²+...+(n-1)²]+ 3[((n-1)+1)(n-1)/2]+ [n-1]

n³-1 -3n(n-1)/2 -[n-1]= 3[1²+2²+3²+4²+5²+...+(n-1)²+n²] - 3n²

[1²+2²+3²+4²+5²+...+(n-1)²+n²]= [n³-1 -3n(n-1)/2 -n+1+ 3n²]/3=

= [2n³ +3n²+n]/6= n[2n² +3n+1]/6=

=n(n+1)(2n+1)/6