# Recursive Formula

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) = a_{0}h(0) + a_{1}h(1) + a_{2}h(2) + … + a_{x – 1}h(x – 1)

Where a_{i }≥ 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

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

There are few recursive formulas to find the n^{th} term based on the pattern of the given data. They are,

- n
^{th}term of Arithmetic Progression a_{n }= a_{n – 1 }+ d for n ≥ 2- n
^{th}term of Geometric Progression a_{n }= a_{n – 1 }× r for n ≥ 2- n
^{th}term in fibonacci Sequence a_{n }= a_{n – 1 }+ a_{n – 2}for n ≥ 2 and a_{0 }= 0 & a_{1 }= 1Where 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 a

_{n }= a_{n-1 }+ da

_{4 }= a_{4-1 }+ d= a

_{3 }+ d= 21 + 10

a

_{4 }= 31Missing 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 a

_{n }= a_{n-1 }+ d

a_{n }= a_{n-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,

a

_{2}/a_{1 }= 3/1 = 3a

_{3}/a_{2 }= 9/3 = 3a

_{5}/a_{4 }= 243/81 = 3Hence ratio between adjacent numbers are same. So given series is in Geometric Progression

Common Ratio (r) = 3

Recursive Function to find nth term is a

_{n }= a_{n-1 }× ra

_{4 }= a_{4-1 }× r= a

_{3 }× r= 9 × 3

a_{4 }= 27Missing 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

a

_{2}/a_{1 }= 4/2 = 2a

_{3}/a_{2}= 8/4 = 2a

_{4}/a_{3}= 16/8 = 2Common Ratio (r) = 2

The given number series is in Geometric progression.

So recursive formula a

_{n }= a_{n-1 }× r

a_{n }= a_{n-1 }× 2

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

**Solution:**

Given that number series is in fibonacci series form.

Also given a

_{3 }= 2a

_{4 }= 4Then a

_{5 }= a_{3 }+ a_{4}= 2 + 3

a_{5 }= 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 n

^{th}term is the result of addition between two previous terms i.e., n – 1^{th}& n – 2^{th }termsExample: a

_{3 }= a_{1 }+ a_{2}3 = 1 + 2

Then a

_{8 }= a_{7 }+ a_{6}= 13 + 8

a_{8 }= 21

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