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!
Free Game
ReplyDeleteSteam Keys
Free Steam Keys
Game Giveaway
Game Steam Giveaway
Free Steam Game Giveaway