SEQUENCES

A sequence is an ordered list of numbers, which are called terms. Each term in a sequence is typically defined by a specific rule or formula. Sequences can be finite (with a limited number of terms) or infinite (continuing indefinitely).

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Arithmetic Sequence 

Each term is obtained by adding a constant

 For example 2, 5, 8, 11, (common difference = 3).

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The Arithmetic formula follows 

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an = the n-th term

a1= the first term

d =  the common difference

n =  the term number​

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Example: The first term of an arithmetic sequence is 5, and the common difference is 3. Find the 10th term 

Solution: a1= 5, d =  3, n =  10​

a10 = 5 + (10-1)3 = 32

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Geometric Sequence

 Each term is obtained by multiplying the previous term by a constant (called the common ratio).

For example 3, 6, 12, 24, (common ratio = 2).

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The Geometric formula follows

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an = the n-th term

a1= the first term

r =  the common ratio

n =  the term number

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Example: The first term of a geometric sequence is 4, and the common ratio is 2. Find the 6th term

Solution: a1= 4, r =  2, n = 6

a6 = 4(2)^5 = 128

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Fibonacci Sequence

Each term is the sum of the two preceding terms

For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

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It is used in various fields, including math, biology, and art, to model natural patterns, analyze algorithms, and create visually appealing designs.

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