Recursive vs Explicit Formulas Math Example 3

Follow the full solution, then compare it with the other examples linked below.

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A sequence is defined by

a_1 = 5

a_n = a_{n-1} + 3

. Write the explicit formula and find

a_{20}

1
Identify the pattern: each term increases by 3, so this is an arithmetic sequence with first term $a_1 = 5$ and common difference $d = 3$ .
2
Write the explicit formula: $a_n = a_1 + (n-1)d = 5 + (n-1)(3) = 5 + 3n - 3 = 3n + 2$ .
3
Find $a_{20}$ : $a_{20} = 3(20) + 2 = 60 + 2 = 62$ . Verify: $a_1 = 3(1)+2 = 5$ ✓, $a_2 = 3(2)+2 = 8 = 5+3$ ✓.

Answer

Explicit formula:

a_n = 3n + 2

;

a_{20} = 62

A recursive formula with a constant addend defines an arithmetic sequence. The explicit formula

a_n = a_1 + (n-1)d

lets you jump directly to any term without computing all previous terms — far more efficient for large

n

About Recursive vs Explicit Formulas

Two ways to define a sequence: recursive uses the previous term(s), explicit gives the $n$ th term directly as a function of $n$ .

A sequence is defined by [formula], [formula]. Find an explicit formula and compute [formula].

A sequence satisfies [formula], [formula]. Find the explicit formula and identify the growth type.

Write a recursive formula for [formula]

Convert [formula] to a recursive formula.