Recursive Formula, Arithmetic, Geometric, Fibonacci Sequence

Recursive Formula: Before exploring the recursive formula, it's essential to revisit the concept of a recursive function. A recursive function defines each term within a sequence based on a previously known term. In other words, it establishes the next term by relying on one or more known preceding terms.

A recursive function, represented as h(x), can be articulated as:

h(x)=a ₀ ​ ⋅h(0)+a ₁ ​ ⋅h(1)+⋯+a _{x−1 ​} ⋅h(x−1)

where a _i ​ ≥0 and at least one of the a _i ​ is greater than 0.

Understanding Recursive Formulas

A recursive formula defines each term within a sequence based on preceding terms. It covers two critical parameters: The initial term of the sequence. The pattern rule that derives any term based on its preceding term(s). Recursive Formulas Various types of sequences possess distinct recursive formulas, outlined as follows:

Recursive Formula for Arithmetic Sequences

The formula to determine the nth term of an arithmetic sequence is:

a _n ​ =a _n−1 ​ +d for n≥2

Where: a n ​ represents the nth term of an arithmetic progression (A.P.).  d signifies the common difference.

Recursive Formula for Geometric Sequences

The formula to find the nth term of a geometric sequence is:

a _n ​ =a _n−1 ​ ⋅r for n≥2

Where: a n ​ represents the nth term of a geometric progression (G.P.).

r denotes the common ratio.

Recursive Formula for Fibonacci Sequences

The formula to discover the nth term of a Fibonacci sequence is:

a _n ​ =a _n−1 ​ +a _n−2 ​ for n≥2

Where: a ₀ ​ =1 and a ₁ ​ =1.

a _n ​ signifies the nth term within the sequence. The practical applications of these recursive formulas will be explored in the subsequent section.

Recursive Formula Solved Examples

Example 1: Given the recursive formula f(x)=5f(x−2)+3 and f(0)=0, finding the value of f(8).

Solution: Starting with the formula

f(8)=5f(6)+3,

f(6)=5f(4)+3, and

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

Given f(0)=0, it follows that:

f(2)=5f(0)+3=3,

f(4)=5×3+3=18,

f(6)=5×18+3=93, and

f(8)=5×93+3=468.

Answer: The value of f(8) is 468.

Example 2: Finding the recursive formula for the arithmetic sequence: 1, 6, 11, 16 .....

Solution: Let a n ​ be the nth term of the sequence, and  d be the common difference. d=a 2 ​ −a 1 ​ =6−1=5

Thus, the  formula is

a _n ​ =a _n−1 ​ +5.

Answer: The  formula for this sequence is

a _n ​ =a _n−1 ​ +5.

Example 3: Given the 13th and 14th terms of the Fibonacci sequence as 144 and 233 respectively, finding the 15th term. Solution:

Using the recursive formula for the Fibonacci sequence,

The 15th term is the sum of the 13th term and the 14th term.

15th term = =13th term + +14th term =

144 + 233 =144+233 = 377

=377

Answer: The 15th term of the Fibonacci sequence is 377.

A recursive formula defines a sequence by expressing each term based on prior terms. It uses established rules to compute subsequent terms, relying on previous values within the sequence, common in arithmetic, geometric, and Fibonacci sequences for determining patterns and behaviors in mathematics and beyond.

Understanding this formulas is crucial in defining sequences by establishing terms based on preceding known terms. Recursive functions, such as those in arithmetic, geometric, and Fibonacci sequences, play a pivotal role in mathematics and real-world applications.

These formulas provide a structured approach to computing the nth term of sequences based on predefined rules. For instance, in arithmetic sequences, the recursive formula defines the next term by adding a constant difference to the previous term. Geometric sequences use a ratio to determine subsequent terms, while Fibonacci sequences add the previous two terms to generate the next term.

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