10.1-3: series, sequences, & sigma notation
how to write a rule for finding the nth term of a sequence
1. find pattern in sequence (ex. +5, etc.). this will be the number attached to n. 2. write multiples of the number found in #1 (ex. if the pattern is +5, write out 5, 10, 15, 20) 3. rewrite the sequence under the multiples 4. find the common difference between both sets of numbers (ex. 5, 10, 15, 20 over 6, 11, 16, 21 shows a common pattern of +1) 5. write the rule → aⁿ = 5n+1
example of recursive sequence
7th term; a₁ = 14; aⁿ = 0.5aⁿ⁻¹ + 3 a₁ = 14 a₂ = 0.5a₁ +3 = 0.5(14) + 3 = 7 + 3 = 10 a₃ = 0.5a₂ +3 = 0.5(10) + 3 = 5 + 3 = 8 etc... until you get to the 7th term.
sequence
a collection that is ordered so that it has a first member, a second member, a third member, etc.; a function whose domain is a set of real numbers
finite sequence
a function whose domain is the first n positive integers only
infinite sequence
a function whose domain is the set of positive integers (1, 2, 3, ...) where aₙⁿ denotes the nth term
arithmetic sequence
a sequence whose consecutive terms have a common difference
geometric sequence
a sequence whose consecutive terms have a common ratio
summation notation
a way to write the sum of the terms of a finite sequence ex. ∑⁵ i=1; 3i = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 45 (basically... i=1 means you start at 1, and the ⁵ means that you go up to 5 as you plug in the 3i, so it's 3(1), 3(2), etc. until you get to 5)
terms of a finite sequence
a₁, a₂, a₃, a₄, ..., aⁿ
terms of the infinite sequence
a₁, a₂, a₃, a₄, ..., aⁿ, ...
arithmetic sequence formula
d = a₂ - a₁
fundamental counting principle
m x n
geometric sequence formula
r = (a₂) / (a₁) * follow PEMDAS when solving formula
permutation
the arrangement of objects in a certain ORDER
combination
the arrangement of objects in which order is NOT important
infinite series
the sum of all the terms of the infinite sequence ex. ∑∞ i=1; aᵢ = a₁ + a₂ + a₃ + ... + aᵢ + ...
find the nth partial sum in a finite series
the sum of the first n terms of an infinite sequence ex. ∑ⁿ i=1; aᵢ = a₁ + a₂ + a₃ + ... + aⁿ
series
the sum of the terms of a finite or infinite sequence finite sequence: 2, 6, 10, 14 finite series: 2 + 6 + 10 + 14 infinite sequence: 1/2, 1/4, 1/8, 1/16, ... infinite series: 1/2 + 1/4 + 1/8 + 1/16 + ...
recursive sequence
･ in a recursive formula, each term is defined as a function of its preceding term(s) - each term is found by doing something to the term(s) immediately in front of them - starting term → a₁ - nth term is an expression of the previous term → aⁿ⁻¹
factorial
･ n! is the product of the positive integers less than n - ex. 5! = 5 x 4 x 3 x 2 x 1 = 120
infinite sequences
･ triangular numbers → 1, 3, 6, 10, 15, 21, 28, ... → ･, ∴, etc. ･ square numbers → 1, 4, 9, 16, 25, 36, 49, ... ･ cubic numbers → 1, 8, 27, 64, 125, ... ･ Fibonacci sequence → 1, 1, 2, 3, 5, 8, 13, ... - formed by adding the two previous terms to get the next one
