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!