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Rule

Formulas for Special Series

When a series has many terms, it can be tedious to calculate its sum. However, there are formulas for special series that simplify calculations. The table shows the formulas for some special series.
Special Series Formula
Sum of terms of
Sum of first positive integers.
Sum of squares of first positive integers.
Sum of cubes of first positive integers.

Proof

Sum of Terms of
Adding the same number times is the same as multiplying the number by In this case, since the number is the result of this operation is which equals

Proof

Sum of First Positive Integers
Mathematical induction will be used to prove the following statement.
The first step in an inductive proof is to show that the equation holds true for
The statement is true for Now, assume that the statement is true for some positive integer
In the final step, the aim is to show that the statement is true for To do so, the above equation will be manipulated using the Properties of Equality.
Simplify right-hand side
The left-hand side of the above equation is the sum of the first positive integers. The right-hand side is the expression obtained when substituting for Therefore, the statement holds true for all positive integers.

Proof

Sum of Squares of First Positive Integers
The following statement will be proved using mathematical induction.
For the first step, it should be shown that the equation holds true for
Evaluate left-hand side
Evaluate right-hand side
The statement is true for Now, assume that the statement is true for some positive integer
Next, show that the statement is true for To do so, start by adding to both sides of the above equation.
Add the terms on the right-hand side, then factor the numerator of the fraction.
Simplify right-hand side
The left-hand side is the sum of the squares of the first positive integers. The right-hand side is the expression obtained when substituting for Therefore, the statement holds for all positive integers

Proof

Sum of Cubes of First Positive Integers
The following statement can be proved by mathematical induction.
In the first step, it should be shown that the equation holds true for
Evaluate left-hand side
Evaluate right-hand side
The statement is true for Now, assume that the statement is true for some positive integer
Finally, to show that the statement is true for the above equation will be manipulated.
Simplify right-hand side

This final equation shows that the statement is true for Therefore, the statement holds for all positive integers