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Sequences

This guide provides an overview of arithmetic sequences, including notes on recursive and explicit equations, and examples of finding the rule for different sequences.

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Sequences

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  1. Sequences Notes Day 1

  2. 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

  3. 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  5 is the first term, must define • t(n+1) = t(n) + 4  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

  4. 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) = 4n + 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

  5. 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 + 4 + 4 = 21 Right? 21 is 5th 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) = 4n +1 4 is generator (what you add) 1 is zero term n is term number

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

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

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