How to Calculate Sequences?

Calculating sequences involves finding the general term (formula) that describes the sequence or determining the value of a specific term in the sequence. The approach depends on the type of sequence. Here's a breakdown:

1. Identifying the Type of Sequence

Before calculating anything, identify the type of sequence you're dealing with. Common types include:

• Arithmetic Sequences: A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.
• Geometric Sequences: A sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio.
• Fibonacci Sequence: Each term is the sum of the two preceding ones.
• Other Sequences: Sequences that don't fall into the above categories may have more complex patterns or be defined recursively.

2. Arithmetic Sequences

Finding the General Term (a_n)

The general term (a_n) of an arithmetic sequence is given by:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term of the sequence
• a₁ is the first term of the sequence
• n is the position of the term in the sequence
• d is the common difference

Example:

Consider the arithmetic sequence: 2, 5, 8, 11, ...

• a₁ = 2 (the first term)
• d = 3 (the common difference, 5-2 = 3, 8-5 = 3, etc.)

Therefore, the general term is:

a_n = 2 + (n - 1)3 = 2 + 3n - 3 = 3n - 1

So, to find the 10th term (a₁₀), substitute n = 10:

a₁₀ = 3(10) - 1 = 29

Finding the Sum of the First 'n' Terms (S_n)

The sum of the first 'n' terms of an arithmetic sequence is given by:

S_n = (n/2) [2a₁ + (n - 1)d]

OR

S_n = (n/2) (a₁ + a_n)

Where:

• S_n is the sum of the first n terms
• n is the number of terms
• a₁ is the first term
• d is the common difference
• a_n is the nth term

Example:

Using the same sequence 2, 5, 8, 11, ... Let's find the sum of the first 5 terms (S₅).

• n = 5
• a₁ = 2
• d = 3

S₅ = (5/2) [2(2) + (5 - 1)3] = (5/2) [4 + 12] = (5/2) * 16 = 40

3. Geometric Sequences

Finding the General Term (a_n)

The general term (a_n) of a geometric sequence is given by:

*a_n = a₁ r^(n-1)**

Where:

• a_n is the nth term
• a₁ is the first term
• r is the common ratio
• n is the term position

Example:

Consider the geometric sequence: 3, 6, 12, 24, ...

• a₁ = 3
• r = 2 (6/3 = 2, 12/6 = 2, etc.)

Therefore, the general term is:

a_n = 3 * 2^(n-1)

To find the 6th term (a₆):

a₆ = 3 2^(6-1) = 3 2⁵ = 3 * 32 = 96

Finding the Sum of the First 'n' Terms (S_n)

The sum of the first 'n' terms of a geometric sequence is given by:

*S_n = a₁ (1 - rⁿ) / (1 - r)** (where r ≠ 1)

Where:

• S_n is the sum of the first n terms
• a₁ is the first term
• r is the common ratio
• n is the number of terms

Example:

Using the sequence 3, 6, 12, 24, ... Let's find the sum of the first 4 terms.

• a₁ = 3
• r = 2
• n = 4

S₄ = 3 (1 - 2⁴) / (1 - 2) = 3 (1 - 16) / (-1) = 3 * (-15) / (-1) = 45

4. Fibonacci Sequence

The Fibonacci sequence is defined recursively:

• F₀ = 0
• F₁ = 1
• F_n = F_n-1 + F_n-2 for n > 1

To calculate a term, you simply add the two preceding terms.

Example:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...

5. Other Sequences

For sequences that don't fit the above patterns, you might need to:

• Look for a pattern: Analyze the differences or ratios between consecutive terms.
• Express the sequence recursively: Define each term in terms of previous terms.
• Use a specific formula: Some sequences have unique formulas that can be used to calculate the terms.

Understanding the type of sequence and applying the correct formula or method is crucial for calculating sequences.