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Showing posts from September, 2015

### 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!