Skip to page content

Geometric Progression — Part 2: Sum of n terms and applications

Beta

Lesson

Geometric Progression, Part 2

Adding the terms of $1, 2, 4, 8, \ldots$ gives $1, 3, 7, 15, \ldots$ : always one less than the next term.

In this lesson you will

Derive and use the sum of the first n terms of a geometric progression
Choose the sum formula that suits r > 1, r < 1 and r = 1
Find how many terms give a total, and a term from two sums
Solve growth, depreciation and doubling problems with geometric progressions

term
running total

For $1, 2, 4, \ldots$ the sum of $n$ terms is $2^{n} - 1$.

Concept

Deriving the sum

Multiply the sum by $r$ and subtract: every term except the first and the last cancels.

S_n = \dfrac{a\left(r^{n} - 1\right)}{r - 1}

Multiply by $r$ and subtract: everything in the middle cancels.
Step	Working
Write the sum	$S_n = a + ar + ar^{2} + \cdots + ar^{n-1}$
Multiply by $r$	$rS_n = ar + ar^{2} + \cdots + ar^{n-1} + ar^{n}$
Subtract	$rS_n - S_n = ar^{n} - a$
Divide by $r - 1$	$S_n = \dfrac{a\left(r^{n} - 1\right)}{r - 1}$