Sequences and Series Review

Vocabulary flashcards covering geometric series, the Fibonacci sequence formula, the golden ratio, harmonic sequences, and arithmetic sequences from the lecture notes.

Finite Geometric Series Sum

The formula for calculating the sum of a finite geometric series, given as $S_n = \frac{a_1 (r^n - 1)}{r - 1}$

Infinite Geometric Series Sum

The formula for calculating the sum of an infinite geometric series, given as $S_\infty = \frac{a_1}{1 - r}$

Fibonacci Sequence Formula

An explicit formula for finding the $n$-th term of the Fibonacci sequence, given as $a_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$

Golden Ratio ($\phi$)

A constant denoted by $\phi$, with the approximate value $\phi \approx 1.618$

Harmonic Sequence

A sequence of numbers whose reciprocals form an arithmetic sequence.

Arithmetic Sequence General Term Formula

The formula for the $n$-th term of an arithmetic sequence, given as $a_n = a_1 + (n - 1)d$

Finite Geometric Series Sum Example

For the series $5 + 15 + 45 + 135$, where $a_1 = 5$, $r = 3$, and $n = 4$: $S_4 = \frac{5(3^4 - 1)}{3 - 1} = \frac{5(81 - 1)}{2} = 200$

Infinite Geometric Series Sum Example

For the infinite series $16 + 8 + 4 + \dots$, where $a_1 = 16$ and $r = \frac{1}{2}$: $S_\infty = \frac{16}{1 - \frac{1}{2}} = \frac{16}{0.5} = 32$

Fibonacci Sequence Formula Example

To find the $2\text{nd}$ term ($n = 2$) using Binet's formula with $\phi \approx 1.618$: $a_2 = \frac{1.618^2 - (1 - 1.618)^2}{\sqrt{5}} = \frac{2.618 - 0.382}{2.236} = 1$

Golden Ratio ($\phi$) Example

The ratio of consecutive Fibonacci terms approaches $\phi$; for example, $\frac{21}{13} \approx 1.615$, which approximates $\phi \approx 1.618$

Harmonic Sequence Example

For the sequence $\{\frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}\}$, taking reciprocals gives the arithmetic sequence $4, 7, 10, 13$ with common difference $d = 3$

Arithmetic Sequence General Term Example

For an arithmetic sequence starting at $a_1 = 7$ with common difference $d = 5$, the $4\text{th}$ term is $a_4 = 7 + (4 - 1) \times 5 = 22$