Evaluating the Sum of a Geometric Sequence

This is a short blog post that shows you an easy and intuitive way to derive the formula for the summation of an infinite geometric sequence. Let \( 0 \leq p < 1 \), and let \( a \) be some constant; then we wish to find the value of \( x \) such that $$ x = \sum_{k=0}^\infty a p^k $$ Writing out the first few terms of the summation, we get: $$ x = a + a p + a p^2 + a p^3 + \dots $$ Rewriting the equation by factoring out a \( p \) from every term except the first, we get: $$ x = a + p (a + a p + a p^2 + \dots) $$ Notice that the expression in parenthesis is exactly how \( x \) is defined. Replacing the expression with \( x \) leaves us with: $$ x = a + p x $$ Solving the equation for \( x \) yields $$ x = \frac{a}{1-p} $$ Just remember this simple derivation and you will never have to look up the formula for evaluating the sum of an infinite geometric sequence ever again!


Popular posts from this blog

Multi-Core Programming with Java

Interpolation Search Explained

Beat the Streak: Day Three