Polynomial Series

Polynomial Series Formulae

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Sum of n values of 1

n

1

Σ

r = 1

=

n
Sum of n constant terms

n

c

Σ

r = 1

=

nc
Sum of the first n natural numbers

n

r

Σ

r = 1

=
n(n+1)
2
Sum of the first n square numbers

n

r2

Σ

r = 1

=
n(n+1)(2n+1)
6
Sum of the first n cubic numbers

n

r3

Σ

r = 1

=
n2(n+1)2
4

Sum of the first n constant terms

rth term: 1

n

c

Σ

r = 1

=

c + c + c + c + … + c

=

nc

Sum of the first n natural numbers

rth term: r

n

r

Σ

r = 1

=

1 + 2 + 3 + 4 + … + n

=

n(n + 1)
2

Derivation of the Formula

Let S

=

n

r

Σ

r = 1

S

=

1

+ 2

+ 3

+ …

+ (n - 1)

+ n

S

=

n

+ (n - 1)

+ (n - 2)

+ …

+ 2

+ 1

2S

=

[1 + n]

+ [2 + (n - 1)]

+ [3 + (n - 2)]

+ …

+ [(n - 1) + 2]

+ [n + 1]

=

(n + 1)

+ (n + 1)

+ (n + 1)

+ …

+ (n + 1)

+ (n + 1)

=

n(n + 1)

S

=
n(n+1)
2

Sum of the first n square numbers

rth term: r2

n

r2

Σ

r = 1

=

1 + 4 + 9 + 16 + … + n2

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

Derivation of the Formula

Using the method of differences

Let S =

n

r2

Σ

r = 1

n

((r + 1)3 - r3)

Σ

r = 1

=

(23 - 13) + (33 - 23) + … + (n3 - (n-1)3) + ((n+1)3 - n3)

=

(n+1)3 - 1

=

n3 + 3n2 + 3n

n

((r + 1)3 - r3)

Σ

r = 1

=

n

(3r2 + 3r + 1)

Σ

r = 1

=

3S +

3n(n+1)
2

+ n

=

3S +

3n(n+1) + 2n
2

n3 + 3n2 + 3n

=

3S +

3n2 + 5n
2

2n3 + 6n2 + 6n

=

6S + 3n2 + 5n

6S

=

2n3 + 6n2 + 6n - 3n2 - 5n

=

2n3 + 3n2 + n

=

n(n+1)(2n+1)

S

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

Sum of the first n cubic numbers

rth term: r3

n

r3

Σ

r = 1

=

1 + 8 + 27 + 64 + … + n3

=
n2(n+1)2
4

Derivation of the Formula

Using the method of differences

Let S =

n

r3

Σ

r = 1

n

((r + 1)4 - r4)

Σ

r = 1

= (24 - 14) + (34 - 24) + … + (n4 - (n-1)4) + ((n+1)4 - n4)

= (n+1)4 - 1

= n4 + 4n3 + 6n2 + 4n

n

((r + 1)4 - r4)

Σ

r = 1

=

n

(4r3 + 6r2 + 4r + 1)

Σ

r = 1

=

4S +

6n(n+1)(2n+1)
6

+
4n(n+1)
2

+ n

=

4S + n(n+1)(2n+1) + 2n(n+1) + n

=

4S + 2n3 + 5n2 + 4n

n4 + 4n3 + 6n2 + 4n

=

4S + 2n3 + 5n2 + 4n

4S

=

n4 + 2n3 + n2

=

n2(n + 1)2

S

=
n2(n+1)2
4

Polynomials Series

Generate an expression for the sum of the first n terms of a polynomial series containing powers from 0 to 3.

Expression:

n

ar3 + br2 + cr + d

Σ

r = 1

a:

b:

c:

d:

Calculate the sum for specific values of p and q, starting at r = p and continuing to r = q. Note that if p is greater than or equal to q, only the expression will be generated. p is also limited to values greater than or equal to 0.

Calculate:

q

ar3 + br2 + cr + d

Σ

r = p

p:

q:

Expression

n

(r3 + r2 + r + 1)

Σ

r = 1

=
3n4 + 10n3 + 15n2 + 20n
12

=
n(3n3 + 10n2 + 15n + 20)
12

Calculation

5

(r3 + r2 + r + 1)

Σ

r = 1

=

 300

Formulae Used

Sum of n constant values of 1

n

1

Σ

r = 1

=

n
Sum of the first n numbers

n

r

Σ

r = 1

=
n(n+1)
2
Sum of the first n square numbers

n

r2

Σ

r = 1

=
n(n+1)(2n+1)
6
Sum of the first n cube numbers

n

r3

Σ

r = 1

=
n2(n+1)2
4

Workings for Expression

n

(r3 + r2 + r + 1)

Σ

r = 1

=

n

r3

Σ

r = 1

 +

n

r2

Σ

r = 1

 +

n

r

Σ

r = 1

 +

n

1

Σ

r = 1

=
n2(n+1)2
4
 +
n(n+1)(2n+1)
6
 +
n(n+1)
2
 +

n

=
n(3n3 + 10n2 + 15n + 20)
12