Explicit formulas Sequences: Lesson 3

Explicit formulas An Explicit Formula is a formula for calculating each term using the index n (the term number in the sequence) An explicit formula allows you to find any element of a sequence without knowing the element before it. Plug in the term number you are trying to find as n, input, and simplify to find an , output. Like plugging an input into a function. (we have already practiced using explicit formulas!) Explicit formulas can be used for arithmetic or geometric sequences

Use Explicit formulas for arithmetic sequences: Use the rule an = 2+n Find a10 (the 10th term) Plus in 10 as n and evaluate a10 = 2 + (10) = 12 So the 10th term is 12 a10 = 12 Write the first 5 terms of this arithmetic sequence. xn = 19 - 6(n-1) Substitute 1 in for “n.” and evaluate x1 = 19 - 6(1-1) = 19 - 6(0) = 19 x2 = 19 - 6(2-1) = 19 - 6 = 13 and then repeat for n=3 thru 5 19, 13, 7, 1, -5, …

Use Explicit formulas for geometric sequences: Write the first 5 terms of this geometric sequence. xn = (3)(2)(n-1) Substitute 1 in for “n.” and evaluate x1 = (3)(2)(1-1) x1 = (3)(2)(0) x1 = (3)(1) = 3 x2 = (3)(2)(2-1) = (3)(2)(1) = 6 and then repeat for n=3 thru 5 Use the rule an = 2n Find a5 (the 5th term) Plus in 5 as n and evaluate a5 = 2(5) = 32 So the 5th term is 32 a5 = 32 3, 6, 12, 24, 48, …

Write Explicit formulas for Arithmetic Sequences Arithmetic sequences work like this: a, a + d, a + 2d, a + 3d, … a or a1 is the first term. d is the common difference between terms To write an arithmetic sequence, the rule is: an = a1 + d(n-1) (We use “n-1” because d is not used on the 1st term.)

Write Explicit formulas for Arithmetic Sequences Write the explicit rule for the sequence 19, 13, 7, 1, -5, … Start with the formula: an = a + d(n-1) a is the first term = 19 d is the common difference: -6 The rule is: an = 19 - 6(n-1)

Write Explicit formulas for Arithmetic Sequences Write the explicit rule for the sequence 4, 7, 10, 13, 16 Start with the formula: an = a + d(n-1) a is the first term = 4 d is the common difference: +3 The rule is: an = 4 + 3(n-1) Sequence Term Term a1 4 a2 7 a3 10 a4 13 a5 16 Notice, this is also the linear function y=3x+1. The 3 is the rate of change but the y-intercept is 1 not 4 because the linear function starts when x=0 and a sequence starts at a1 similar to x=1.

Write Explicit formulas for geometric Sequences Geometric sequences work like this: a, ar, ar2, ar3, … a (or a1) is the first term r is the common ratio between terms To write a geometric sequence, the rule is: an = a1r(n-1) (We use “n-1” because r is not used on the 1st term, so ar0 is the 1st term.)

Write Explicit formulas for geometric Sequences Write the explicit rule for the sequence 3, 6, 12, 24, 48, … Start with the formula: an = ar(n-1) a is the first term = 3 r is the common ratio: 2 The rule is: an = (3)(2)(n-1) (Order of operations states that we would take care of exponents before you multiply.)

Practice 1. Find the 12th term of this sequence. xn = 19 - 6(n-1) 3. Write an explicit formula for the sequence 9, 1, -7, -15… x12 = -47 x12 = 6,144 an = 9 – 8(n-1)

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