Exponents, Geometric Series, Fibonacci Sequence, and Harmonic Sequences

Exponent Rules and Algebraic Expressions

• Definition and conversion rules for negative exponents and rational exponents:

  • Division rule of powers with identical base:

$\frac{x^0}{x^1} = x^{0-1} = x^{-1}$

• Requirement for exponents in polynomial expressions: exponents must be positive integers.

• Radical form conversion to exponential form:

$\sqrt{x} = x^{\frac{1}{2}}$

- The power of $x$ under the radical ($1$) serves as the numerator.
- The index of the root ($2$) serves as the denominator.

• Classification and analysis of algebraic expressions:

  • $3x^{-1}$: Contains a negative exponent ($-1$); therefore, it is not a polynomial term.

  • $x^2 + 4x - 1$: A quadratic polynomial with positive integer exponents ($2$ and $1$).

  • $x^{-3} + 5$: Contains a negative exponent ($-3$).

  • $\sqrt{x} + 2$: Equivalent to $x^{\frac{1}{2}} + 2$, containing a fractional exponent ($\frac{1}{2}$).

  • Polynomial term combination:

$\frac{1}{3}x^2 + \frac{1}{2}x - \frac{1}{4}$

Geometric Series

• Finite Geometric Series:

  • A geometric series is the sum of the terms of a geometric sequence.

  • Formula for the sum $S_n$ of the first $n$ terms of a finite geometric series:

$S_n = \frac{a_1(r^n - 1)}{r - 1}$

- $S_n$ represents the finite sum.
- $a_1$ represents the first term of the sequence.
- $r$ represents the common ratio.
- $n$ represents the total number of terms.

• Infinite Geometric Series:

  • Formula for the sum $S_\infty$ of an infinite geometric series:

$S_\infty = \frac{a_1}{1 - r}$

- $S_\infty$ represents the infinite sum.
- $a_1$ represents the first term.
- $r$ represents the common ratio.

Fibonacci Sequence

• Definition: A sequence where each term is generated by adding the two preceding terms.

• Explicit formula (Binet's Formula) for the $n$\text{-th} term $a_n$:

$a_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$

• Golden Ratio (represented by $\phi$):

$\phi = 1.618$

Harmonic Sequence

• Definition: A sequence formed by taking the reciprocals of the terms of an arithmetic sequence.

• Formula for the $n$\text{-th} term of an arithmetic sequence:

$a_n = a_1 + (n - 1)d$

• $a_n$ is the $n$\text{-th} term of the arithmetic sequence.

• $a_1$ is the first term.

• $d$ is the common difference.

• $n$ is the term position.

  • Example 1:

• Harmonic sequence:

$\{\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{10}, \frac{1}{12}, \frac{1}{14}, \frac{1}{16}\}$

• Corresponding underlying arithmetic sequence of denominators:

$2, 4, 6, 8, 10, 12, 14, 16$

- The terms of the denominator sequence have a common difference of $d = 2$

• Example 2:

  • Harmonic sequence:

$2, 1, \frac{2}{3}, \frac{1}{2}, \frac{2}{5}, \frac{1}{3}, \frac{2}{7}$

• Sequence of reciprocals forming an arithmetic sequence:

$\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3, \frac{7}{2}$

• Rewritten with common denominators:

$\frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{5}{2}, \frac{6}{2}, \frac{7}{2}$

- The common difference for this underlying arithmetic sequence is $d = \frac{1}{2}$

• Fractional Operations and Common Difference Calculations:

  • Difference calculation between harmonic terms:

$\frac{1}{10} - \frac{1}{8} = \frac{8 - 10}{80} = \frac{-2}{80} = \frac{-1}{40}$

• Alternative difference calculation between harmonic terms:

$\frac{1}{8} - \frac{1}{6} = \frac{3 - 4}{24} = \frac{-1}{24}$

$\frac{1}{8} - \frac{1}{6} = \frac{6 - 8}{48} = \frac{-2}{48} = \frac{-1}{24}$

• Common difference verification for arithmetic terms:

$2 - \frac{3}{2} = \frac{4 - 3}{2} = \frac{1}{2}$

$\frac{3}{2} - 1 = \frac{3 - 2}{2} = \frac{1}{2}$

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