Sum of Geometric Sequence/Proof 2
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Theorem
Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$.
Let $n \in \N_{>0}$.
Then:
- $\ds \sum_{j \mathop = 0}^{n - 1} x^j = \frac {x^n - 1} {x - 1}$
Proof
Let $\ds S_n = \sum_{j \mathop = 0}^{n - 1} x^j$.
Then:
\(\ds \paren {x - 1} S_n\) | \(=\) | \(\ds x S_n - S_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \sum_{j \mathop = 0}^{n - 1} x^j - \sum_{j \mathop = 0}^{n - 1} x^j\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^n x^j - \sum_{j \mathop = 0}^{n - 1} x^j\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^n + \sum_{j \mathop = 1}^{n - 1} x^j - \paren {x^0 + \sum_{j \mathop = 1}^{n - 1} x^j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^n - x^0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^n - 1\) |
The result follows.
$\blacksquare$
Sources
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: Exercise $1.2: 1$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes: Footnote $2$