12 2 12 5 Arithmetic Sequences Arithmetic Pattern
- Slides: 8
12. 2 & 12. 5 – Arithmetic Sequences • Arithmetic : Pattern is ADD or SUBTRACT same number each time. • d = common difference – If add: d positive – If subtract: d negative • Explicit Formula a = a₁ + (n-1)(d) – Must know FIRST TERM & DIFFERENCE – SIMPLIFY • Distribute • Combine like terms
Examples • 1. Are these sequences ARITHMETIC? If yes, what is the d? • A) 1, -2, -5, -8, -11, … • B) 3, 6, 12, 24, … • C) 5, 7, 9, 11, 13, …
Examples: • 2. Write a rule (explicit formula) for the arithmetic sequence. Then find a₂₀. • A) 5, 11, 17, 23, 29, … • B) 20, 15, 10, 5, …
Explicit formula if unknown info • If either (or both) the FIRST TERM & the DIFFERENCE are unknown, then a variation of the explicit formula is used. (once for each unknown – then traditional formula is used) • Later = sooner + (subtract subscripts)(d) – Find d – Use found d, unknown first term & a given term to find a₁ – Use found d and found a₁ in traditional explicit formula and simplify.
Example: • 3. Write the rule (explicit formula) for the nth term, then graph first 6 terms. a₄=96, d=-14
Example • Write the rule (explicit) for the nth term. • a₆=39 and a₁₄=79
Recursive Formula • 2 Parts: a₁ = # and a = a + # • Ex: Write recursive formula(rule): 1, -2, -5, -8, -11 • Ex: Write recursive formula(rule): 7, 9, 11, 13 • Ex: Write the first 5 terms: a₁=5, a = a +4
Arithmetic Sum (SERIES) • To find the SUM of a finite number or terms • S = (n/2) (starting value + ending value) • Find the SUM of the arithmetic series (-9+11 n) • Find the SUM of the arithmetic series (-3 n +10)