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Arithmetic Sequence Recursive Formula

Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. Recursion requires that you know the value of the term immediately before the term you are trying to find.

Speaking about the Arithmetic Sequence Recursive Formula, it has two parts: first a starting value that begins the sequence and a recursion equation that shows how terms of the sequence related to the preceding terms.

Here is the formula of Arithmetic Sequence Recursive: $t_{n}$ = $t_{n-1}$

Arithmetic Sequence Recursive formula may list the first two or more terms as starting values depending upon the nature of the sequence. However, the $a_{n}$ portion of the is also dependent upon the previous two or more terms in the sequence.

Here are few examples:

Example 1: Write the first four terms of the sequence when:  $a_{1}$ = – 4 and $a_{n}$ = $a_{n-1}$ + 5
In recursive formulas, each term is used to produce the next term.  Follow the movement of the terms through the steps given below.
Given:
$a_{1}$
= – 4
And
$a_{n}$ = $a_{n-1}$ + 5 (each term is 5 more than the term before)

Solution:
n = 2
$a_{2}$ = $a_{2-1}$ + 5
$a_{2}$ = -4 +5
$a_{2}$ = 1

n = 3
$a_{3}$ = $a_{3-1}$ + 5
$a_{3}$ = 1 + 5
$a_{3}$ = 6

n = 4
$a_{4}$ = $a_{4-1}$ + 5
$a_{4}$ = 6 + 5
$a_{4}$ = 11

Answer:  -4, 1, 6, 11

Example 2: Find the recursive formula when the sequence  2, 4, 6, 8, 10….
Considering this sequence, it can be represented in more than one manner. The given sequence can be represented as either an explicit (general) formula or a recursive formula.
Explicit Formula: $a_{n}$ = $2_{n}$
Recursive Formula: $a_{1}$ = 2 and $a_{n}$ = $a_{n-1}$ + 2