# Recursive Formula

• Last Updated : 22 Mar, 2022

Recursion can be defined by two properties. A base case and recursion step. The base case is a terminating scenario that doesn’t use recursion to produce results. The recursion step consists of a set of rules that reduces the successive cases to forward to the base case.

### Recursive Formula

A recursive function is a function that defines each term of a sequence using the previous term i.e., The next term is dependent on the one or more known previous terms. Recursive function h(x) is written as-

h(x) = a0h(0) + a1h(1) + a2h(2) + … + ax – 1h(x – 1)

Where ai ≥ 0 and i = 0, 1, 2, 3, … x – 1

Recursive Formula is a formula that defines the each term of sequence using the previous/preceding terms. It defines the following parameters

1. The first term of the sequence
2. The pattern rule to get any term from its previous terms.

There are few recursive formulas to find the nth term based on the pattern of the given data. They are,

• nth term of Arithmetic Progression an = an – 1 + d for n ≥ 2
• nth term of Geometric Progression an = an – 1 × r for n ≥ 2
• nth term in fibonacci Sequence an = an – 1 + an – 2 for n ≥ 2 and a0 = 0 & a1 = 1

Where d is common difference and r is the common ratio

### Sample Problems

Question 1: Given a series of numbers with a missing number in middle 1, 11, 21, ?, 41. Using recursive formula find the missing term.

Solution:

Given,

1, 11, 21, _, 41

First term (a) = 1

Difference between terms = 11 – 1 = 10

21 – 11 = 10

So the difference between numbers is same.

Common Difference (d) = 10

Recursive Function to find nth term is an = an-1 + d

a4 = a4-1 + d

= a3 + d

= 21 + 10

a4 = 31

Missing term in the given series is 31.

Question 2: Given series of numbers 5, 9, 13, 17, 21,… From the given series find the recursive formula

Solution:

Given number series

5, 9, 13, 17, 21,…

first term (a) = 5

Difference between terms = 9 – 5 = 4

13 – 9 = 4

17 – 13 = 4

21 – 17 = 4

So the difference between numbers is same.

Common Difference (d) = 4

The given number series is in Arithmetic progression.

So recursive formula an = an-1 + d

an = an-1 + 4

Question 3: Given a series of numbers with a missing number in middle 1, 3, 9, _, 81, 243. Using recursive formula find the missing term.

Solution:

Given,

1, 3, 9, _, 81, 243

First term (a) = 1

The difference between numbers are large so It should not be in Arithmetic progression.

Let’s check whether it is in Geometric progression or not,

a2/a1 = 3/1 = 3

a3/a2 = 9/3 = 3

a5/a4 = 243/81 = 3

Hence ratio between adjacent numbers are same. So given series is in Geometric Progression

Common Ratio (r) = 3

Recursive Function to find nth term is an = an-1 × r

a4 = a4-1 × r

= a3 × r

= 9 × 3

a4 = 27

Missing term in the given series is 27.

Question 4: Given series of numbers 2, 4, 8, 16, 32, … From the given series find the recursive formula.

Solution:

Given number series,

2, 4, 8, 16, 32, …

First term (a) = 2

Difference between terms = 4 – 2 = 2

8 – 4 = 4

It is not in A.P as difference between numbers are not same.

Let’s check whether it is in Geometric progression or not

a2/a1 = 4/2 = 2

a3/a2 = 8/4 = 2

a4/a3 = 16/8 = 2

Common Ratio (r) = 2

The given number series is in Geometric progression.

So recursive formula an = an-1 × r

an = an-1 × 2

Question 5: Find the 5th term in a Fibonacci series if the 3rd and 4th terms are 2,3 respectively.

Solution:

Given that number series is in fibonacci series form.

Also given a3 = 2

a4 = 4

Then a5 = a3 + a4

= 2 + 3

a5 = 5

Question 6: Find the next term in the given series 1, 1, 2, 3, 5, 8, 13, …

Solution:

Given,

1, 1, 2, 3, 5, 8, 13,…

The given series is in fibonacci form because every nth term is the result of addition between two previous terms i.e., n – 1th & n – 2th terms

Example: a3 = a1 + a2

3 = 1 + 2

Then a8 = a7 + a6

= 13 + 8

a8 = 21

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