Sequences Notes Day 1 Arithmetic Sequences Sequences that

Sequences Notes Day 1

Arithmetic Sequences • Sequences that grow by adding or subtracting (which is adding a negative #) a common difference • Common difference is how the sequence grows consistently, also known as the generator • ex) 5, 9, 13, 17, … adding 4 • To define a sequence, use either recursive or explicit equations

Recursive Eqn • Used to find the next term. • Requires one defined term in the sequence & the generator • ex) 5, 9, 13, 17, … • t(1) = 5 • t(n+1) = t(n) + 4 5 is the first term, must define this defines how you get to your next term (adding 4) • Must have both pieces to be a complete recursive eqn • Note: • t(1) means first term • t(n) means current term • t(n+1) means next term • n means term number

Explicit Eqn • Used to find any term in the sequence • Also known as the rule of the sequence • Requires knowing the generator & zero term • ex) 5, 9, 13, 17, … • t(n) = 4 n + 1 where it comes from: • 5, 9, 13, 17, … 5 is first term, so to find zero term, go back one • Generator is adding 4, so subtract 4 to find zero term • 5– 4=1 1 is zero term

• So: • 1, 5, 9, 13, 17, … is sequence beginning w/ zero term • 1+4=5 5+4=9 9 + 4 = 13 13 + 4 = 17 17 + 4 = 21 …. • So 1 + 4 + 4 + 4 = 21 Right? 21 is 5 th term • Or 1 + 4(5) = 21 Right? Repeated addition rewritten as multiplication • 5 is also the term number (n) for 21 • So equation t(n) = 4 n +1 4 is generator (what you add) 1 is zero term n is term number

Defined Formulas • Recursive: • Explicit: • t(1) = first term • t(n+1) = t(n) + generator • t(n) = m n + b • Where: • • • Note: Red areas are the only part that you change in eqn! m = generator b = zero term/ starting point t(n) = term number

Try for yourself! • Find the rule (explicit eqn) for each sequence below 1. 11, 20, 29, 38, … 2. 30, 25, 20, 15, … 3. 15, 12, 9, 6, … 4. -7, -1, 5, 11, … • Remember, explicit eqn or RULE is in the form t(n) = m n + b