Telescoping Sums & Mathematical Induction

Posted: March 9, 2016 in Calculus: An Introduction, Mathematical Induction, Telescoping Sums
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As the title indicates, telescoping sums and mathematical induction are the focus of this entry. Having said that, since the formula for sum of triangular numbers is required in the telescoping sum chosen, I’ve included one version of its derivation here as well.

The method shown below for the sum of triangular numbers involves duplicating the original “stack of blocks”; this duplicate stack is then rotated and affixed to the original stack to form a rectangle. Since two identical stacks have been combined to form this rectangle, its area must be halved to arrive at the desired result.

 

Sum of Triangular Numbers

Sum of Triangular Numbers2

Click on the link provided here to complete the rectangle shown above.

 

Mathematical Induction

When using induction to prove a mathematical statement to be true, the statement itself must first be given. A “base case” is shown initially to prove that the statement holds true for our starting point; this is usually either “0” or “1”.

The inductive proof for the sum formula of triangular numbers follows here. Once the statement is proven true for the basis, it is assumed that a value “k” will also hold true. If it can be shown that the statement also holds for a value “k+1”, then it can be assumed that it will hold true for all values of  k≥1.

 

Induction Proof (Sum of Triangular Numbers)

Induction (Sum of Triangular Numbers)

 

 

The formula for sum of squares is derived directly below using telescoping sums. Mathematical induction follows to prove that this formula holds true for all values of the variable.

 

Telescoping Sum (Sum of Squares Formula)

Telescoping Sum

 

Proof by Induction (Sum of Squares)

Induction

Induction2

Other methods are available for deriving each of the sum formulas shown here; another such entry (using linear combinations) will eventually be linked here. Regardless of how each formula is derived, mathematical induction is a very useful tool to test its reliability.

 

Thanks for reading.

 

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