Series and Sequences - Exam Notes

Arithmetic Series

  • Sum of the first n terms: S_n

  • Formula: Sn = \frac{n}{2}[2a1 + (n-1)d] or Sn = \frac{n}{2}(a1 + a_n) when the first and last terms are known.

  • Partial Sum: The sum of a finite number of terms.

Geometric Series

  • Sum of the first n terms: S_n

  • Finite Geometric Series: Sn = a1 \frac{(1 - r^n)}{(1 - r)}, where r \neq 1

Infinite Geometric Series

  • Divergent: If |r| > 1, the sum is too large to be evaluated.

  • Convergent: If |r| < 1, the series approaches a particular number as terms increase.

  • Formula: S = \frac{a_1}{(1 - r)}

Sigma Notation

  • Represents the sum of a series.

  • Form: \sum_{x=1}^{n} (expression)

  • n: stopping point

  • x: variable

Pascal's Triangle

  • Diagonals: 1st diagonal is all 1s, 2nd is counting numbers, 3rd is triangular numbers, 4th is tetrahedral numbers.

  • Symmetrical.

  • Horizontal Sums: Powers of 2.

  • Finding nth element of rth row: \binom{n}{r} = \frac{n!}{(n-r)! r!}

Binomial Theorem

  • Expansion of (a + b)^n

  • There are n + 1 terms.

  • The exponent of a decreases, and the exponent of b increases in each term.

  • Sum of exponents in each term = n

  • Formula for finding a specific term: \binom{n}{r} a^{n-r} b^r

  • To find the coefficient of a specific term, use the binomial coefficient formula.