Math 1 · Unit 9 · Lesson 4 · part 3 of 5
Translating, in both directions
Recursive to explicit , in three steps.
Translating, in both directions
Recursive to explicit, in three steps.
- 1Read the starting term.Find a1 in the rule.
- 2Adds or multiplies?If the rule adds a number, it is arithmetic and that number is d. If it multiplies by a number, it is geometric and that number is r.
- 3Write the explicit rule.Arithmetic: an = a1 + d(n − 1). Geometric: an = a1 · rn − 1.
an + 1 = an + 12, a1 = 4
1. Start: 4.
2. It adds 12: arithmetic, d = 12.
3. an = 4 + 12(n − 1).
Simplify: 4 + 12n − 12 = 12n − 8. The slope is 12 and the y-intercept is −8, not 4. (The zeroth term: 4 − 12 = −8.)
an + 1 = 3 · an, a1 = 5
1. Start: 5.
2. It multiplies by 3: geometric, r = 3.
3. an = 5 · 3n − 1.
Check: n = 1 gives 5 · 30 = 5. n = 2 gives 15. The list is 5, 15, 45, ...
Explicit to recursive, also in three steps.
- 1Put in n = 1.That gives the starting term, a1.
- 2Read d or r from the rule.In a linear rule, d is the number multiplying n (the slope). In an exponential rule, r is the base of the power.
- 3Write the start and the next-from-now rule.a1 = the start, and an + 1 = an + d, or an + 1 = r · an.
an = 3n + 2
1. n = 1: 3 + 2 = 5. So a1 = 5.
2. The number multiplying n is 3: d = 3.
3. a1 = 5, an + 1 = an + 3.
Careful: the start is 5, not 2. The 2 is the zeroth term.
an = 6 · 2n − 1
1. n = 1: 6 · 20 = 6. So a1 = 6.
2. The base is 2: r = 2.
3. a1 = 6, an + 1 = 2 · an.