If t_{1}, t_{2}, t_{3},….,t_{n},… is a set of series or a sequence. Then a recursive formula for this sequence will be needed to compute all the previous terms and find the value of t_{n}.

t_{n} = t_{n-1} |

This formula can also be defined as Arithmetic Sequence Recursive Formula. As you can observe from the sequence itself, it is an **arithmetic sequence**, which includes the first term followed by other terms and a common difference, d between each term is the number you add or subtract to them.

A recursive function can also be defined for a **geometric sequence**, where the terms in the sequence have a common factor or common ratio between them. And it can be written as;

t_{n} = r x t_{n-1} |

## Recursive Formula Examples

Example 1:

Let t_{1}=10 and t_{n}= 2t_{n-1}+1

So the series becomes;

t_{1}=10

t_{2}=2t_{1}+1=21

t_{3}=2t_{2}+1= 43

And so on…

**Example 2: Find the recursive formula which can be defined for the following sequence for n > 1.**

65, 50, 35, 20,….

Solution:

Given sequence is 65, 50, 35, 20,….

a_{1} = 65

a_{2} = 50

a_{3} = 35

a_{2} – a_{1} = 50 – 65 = -15

a_{3} – a_{2} = 35 – 50 = -15

Thus, a_{2} = a_{1} – 15

Similarly, a_{3} – a_{2} = -15

From this we can write the recursive formula as: a_{n} = a_{n-1} – 15

**Example 3: Calculate f(9) for the recursive series f(x) = 3. f(x – 2) + 4 which has a seed value of f(3) = 9.**

Solution:

Given,

f(3) = 9

f(x) = 3.f(x – 2) + 4

f(9) = 3.f(9-2) + 4

= 3.f(7) + 5

f(7) = 3.f(7-2) + 4

= 3.f(5) + 4

f(5) = 3.f(5-2) + 4

= 3.f(3) + 4

Substituting f(3) = 9,

f(5) = 3(9) + 4 = 27 + 4 = 31

f(7) = 3(31) + 4 = 93 + 4 = 97

f(9) = 3(97) + 4 = 291 + 4 = 295

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