Math Notes about Sum Formulas

I’m working on enhancing my skills, expanding my knowledge and increasing my chances to pass technical interviews. Mathematics plays an important role in problem-solving. Hence, this notes! This blog post is about Sum Formulas.

Each sum of the form: $$ \sum_{x=1}^{n} x^{k} = 1^{k} + 2^{k} + 3^{k} + .. + n^{k} $$ where k is a positive integer, has a closed-form formula that is a polynomial of degree (k + 1).

Triangular number

\[\sum_{x=1}^{n} x = 1 + 2 + 3 + ... + n = \frac{n (n + 1)}{2}\]

Square pyramidal number

\[\sum_{x=1}^{n} x^{2} = 1^{2} + 2^{2} + 3^{2} + ... + n^{2} = \frac{n (n + 1) (2n + 1)}{6}\]

There is a general formula for such sums, called Faulhaber’s Formula.

In the next lines, I’ll use mathematical induction to proof the Triangular number and the Square pyramidal number formulas.

Triangular number.

$$ \sum_{x=1}^{n} x = 1 + 2 + 3 + ... + n = \frac{n (n + 1)}{2} $$

Base step: Base case, for n = 1

\[L.H.S = 1\] \[R.H.S = \frac{n (n + 1)}{2} = \frac{1 ( 1 + 1)}{2} = \frac{2}{2} = 1\] \[L.H.S = R.H.S = 1\]

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Inductive hypothesis Step The equation is true for an arbitrary integer k

\[1 + 2 + 3 + ... + k = \frac{k(k+1)}{2}\]

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Induction Step

Based on the inductive hypothesis, let’s proof the equation if true for n = k + 1

\[L.H.S = 1 + 2 + 3 ... + k + k + 1\] \[= \frac{k(k+1)}{2} + (k + 1)\] \[= \frac{k(k+1)}{2} + \frac{2(k + 1)}{2}\] \[= \frac{k^2 + k + 2k + 2}{2}\] \[= \frac{k^2 + 3k + 2}{2}\] \[= \frac{(k+1)(k+2)}{2}\] \[= \frac{(k + 1)(k+1+1)}{2}\]

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For n = k + 1

\[R.H.S = \frac{n(n + 1)}{2} = \frac{(k+1)(K + 1 + 1)}{2}\] \[L.H.S = R.H.S \#\]

Square pyramidal number

$$ \sum_{x=1}^{n} x^{2} = 1^{2} + 2^{2} + 3^{2} + ... + n^{2} = \frac{n (n + 1) (2n + 1)}{6} $$

Base step: Base case, for n = 1

\[L.H.S = 1^2 = 1\] \[R.H.S = \frac{1(1+1)(2 \times 1 + 1)}{6} = \frac{2 \times 3}{6} = 1\] \[L.H.S = R.H.S\]

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Inductive hypothesis Step The equation is true for an arbitrary integer k

\[1^2 + 2^2 + 3^2 + ... + k^2 = \frac{k(k+1)(2k+1)}{6}\]

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Induction Step

Based on the inductive hypothesis, let’s proof the equation if true for n = k + 1

\[L.H.S = 1^2 + 2^2 + 3^2 + ... + k^2 + (k+1)^2\] \[= \frac{k(k+1)(2k+1)}{6} + (k+1)^2\] \[= \frac{k(k+1)(2k+1)}{6} + \frac{6(k+1)^2}{6}\]

Given the R.H.S:

\[R.H.S = \frac{n(n+1)(2n+1)}{6} = \frac{(k+1)(k+2)(2k+3)}{6}\]

So, we can ignore the 6 in the denominator from both sides for now:

\[{L.H.S}\times{6} = k(k+1)(2k+1) + 6(k+1)^2\] \[= (k^2 + k)(2k+1) + 6(k^2+2k+1)\] \[= 2k^3 + k^2 + 2k^2 + k + 6k^2 + 12k + 6\] \[{L.H.S}\times{6} = 2k^3 + 9k^2 + 13k + 6\]

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\[{R.H.S}\times{6} = (k+1)(k+2)(2k+3)\] \[= (k^2 + 3k + 2)(2k+3)\] \[= 2k^3 + 6k^2 + 3k^2 + 4k + 9k + 6\] \[{R.H.S}\times{6} = 2k^3 + 9k^2 + 13k + 6\]

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\[L.H.S = \frac{2k^3 + 9k^2 + 13k + 6}{6}\] \[R.H.S = \frac{2k^3 + 9k^2 + 13k + 6}{6}\] \[R.H.S = L.H.S \#\]

Number Progressions

A progression, which is also known as a sequence, is nothing but a pattern of numbers. For example, 3, 6, 9, 12 is a progression because there is a pattern observed where every number here is obtained by adding 3 to its previous number.

There are four types of the most common progressions.

Arithmetic Progression

Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive numbers is constant.

For Example: 3, 7, 11, 15, is an AP where the common difference is 4.

For an AP:

• a is the first term.
• l is the last term.
• d is the common difference.
• The n term could be found as follows: \(a_{n} = a + (n - 1) d\)
• We can find the sum as follows: \(S = \frac{n(a + l)}{2}\)

Let’s proof this formula:

The sum of an arithmetic is $$ S = \frac{n(a + l)}{2} $$

Suppose \(a_{1}, a_{2}, a_{3}, ..., a_{n}\) be an arithmetic progression whose first term is a and the common difference is d.

Then:

\[a_{1} = a\] \[a_{2} = a + d\] \[a_{3} = a + 2d\] \[a_{n} = a + (n - 1) d\]

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\[S_{n} = a + (a + d) + (a + 2d) + ... + \{a + (n - 3)d\} + \{a + (n - 2)d\} + \{a + (n - 1)d\} \;\;\;\;\;\;\;\; (i)\]

And with a simple trick of reversing the order:

\[S_{n} = \{a + (n - 1)d\} + \{a + (n - 2)d\} + \{a + (n - 3)d\} + ... + (a + 2d) + (a + d) + a \;\;\;\;\;\;\;\; (ii)\]

by adding \((i)\) and \((ii)\)

Term (i)	Term (ii)	Sum
a	\(a + (n - 1)d\)	\(2a + (n-1)d\)
a + d	\(a + (n - 2)d\)	\(2a + (n-1)d\)
a + 2d	\(a + (n - 3)d\)	\(2a + (n-1)d\)
…	…	…
		\(2S_{n} = n\{2a + (n-1)d\}\)

\[2S_{n} = \{2a + (n-1)d\} + \{2a + (n-1)d\} + ... + \{2a + (n-1)d\}\]

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\[2S_{n} = n\{2a + (n-1)d\}\] \[2S_{n} = n\{a + a + (n-1)d\}\] \[2S_{n} = n(a + l)\] \[S_{n} = \frac{n(a+l)}{2} \#\]

Geometric Progression

Geometric Progression (GP) is a sequence of numbers where the ratio between any two consecutive numbers is constant.

\[S = \frac{l\times k - a}{k - 1}\]

Harmonic Progression

A series of numbers is said to be in harmonic sequence if the reciprocals (the inverse) of all the elements of the sequence form an arithmetic sequence.

For example: \(\sum_{x=1}^{n}{\frac{1}{x}} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}\)

Fibonacci Progression

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1.

Sequence is defined as, \(F_{0} = 0\) and \(F_{1} = 1\) and \(F_{n} = F_{n-1} + F_{n-2}\)