Examples Using Recursive Formula
Example 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
d = T2 – T1 = T3 – T2
d = 11 – 1 = 21 – 11 = 10
Recursive Function in AP an = an-1 + d
a4 = a4-1 + d
a4 = a3 + d
a4 = 21 + 10
a4 = 31
Example 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
d = T2 – T1 = T3 – T2
d = 9 – 5 = 13 – 9 = 4
Recursive Formula for AP an = an-1 + d
an = an-1 + 4
Example 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
a2/a1 = 3/1 = 3
a3/a2 = 9/3 = 3
a5/a4 = 243/81 = 3
Common Ratio (r) = 3
Recursive Function to find nth term in GP an = an-1 × r
a4 = a4-1 × r
a4 = a3 × r
a4 = 9 × 3
a4 = 27
Example 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
a2/a1 = 4/2 = 2
a3/a2 = 8/4 = 2
a4/a3 = 16/8 = 2
Common Ratio (r) = 2
Recursive Formula an = an-1 × r
an = an-1 × 2
Example 5: Find the 5th term in a Fibonacci series if the 3rd and 4th terms are 2,3 respectively.
Solution:
Given,
- a3 = 2
- a4 = 4
Then in Fibonnaci Sequence, a5 = a3 + a4
a5 = 2 + 3
a5 = 5
Recursive Formula
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 the base case.
A recursion or recursive formula is a formula that is used to tell us the next step in any recursion series. In a recursive series, each next term is dependent on the previous one or two terms. In this article, we will learn about, Recursive Formulas or Recursion Formulas, Examples, and others in detail.
Table of Content
- What is a Recursive Function?
- Recursive Formula
- Recursive Formulas For Sequences
- Recursive Formula for Arithmetic Progression
- Recursive Formula for Geometric Progression
- Recursive Formula for Fibonacci Series
- Useful Sequence And Formulas
- Examples Using Recursive Formula
- Practice Question on Recursive Formula