Hw 7.1.1-3 Arithmetic And Geometric Sequences

HW 7.1.1-3: Mastering Arithmetic and Geometric Sequences

This comprehensive guide dives deep into arithmetic and geometric sequences, crucial concepts in algebra and beyond. We'll cover the fundamental definitions, explore key formulas, and work through numerous examples to solidify your understanding. By the end, you'll be confident in identifying, analyzing, and solving problems related to these sequences. This in-depth exploration ensures you're fully prepared for any assessment or further mathematical studies involving arithmetic and geometric progressions.

Understanding Arithmetic Sequences

An arithmetic sequence (also known as an arithmetic progression) is a sequence where the difference between consecutive terms remains constant. This constant difference is called the common difference, often denoted by 'd'.

Key Characteristics:

Constant Difference: The defining feature is the consistent difference between successive terms.
Linear Growth: The terms grow (or decrease) linearly. Graphing an arithmetic sequence results in a straight line.
Formula: The nth term of an arithmetic sequence can be found using the formula: aₙ = a₁ + (n-1)d where:
- aₙ is the nth term
- a₁ is the first term
- n is the term number
- d is the common difference

Example 1: Identifying an Arithmetic Sequence

Consider the sequence: 2, 5, 8, 11, 14...

Let's calculate the difference between consecutive terms:

5 - 2 = 3
8 - 5 = 3
11 - 8 = 3
14 - 11 = 3

The common difference is 3. Therefore, this is an arithmetic sequence.

Example 2: Finding the nth term

Let's find the 10th term (a₁₀) of the sequence from Example 1, where a₁ = 2 and d = 3.

Using the formula: aₙ = a₁ + (n-1)d

a₁₀ = 2 + (10 - 1)3 = 2 + 27 = 29

The 10th term is 29.

Example 3: Finding the common difference

Given that the 3rd term (a₃) of an arithmetic sequence is 11 and the 7th term (a₇) is 27, find the common difference.

We can set up two equations using the formula:

a₃ = a₁ + (3-1)d = 11 a₇ = a₁ + (7-1)d = 27

Subtracting the first equation from the second:

(a₁ + 6d) - (a₁ + 2d) = 27 - 11 4d = 16 d = 4

The common difference is 4.

Sum of an Arithmetic Sequence

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

Sₙ = n/2 [2a₁ + (n-1)d] or equivalently Sₙ = n/2 (a₁ + aₙ)

Example 4: Finding the sum

Find the sum of the first 15 terms of the arithmetic sequence 4, 7, 10, 13...

Here, a₁ = 4, d = 3, and n = 15. Using the first formula:

S₁₅ = 15/2 [2(4) + (15-1)3] = 15/2 [8 + 42] = 15/2 (50) = 375

Alternatively, using the second formula, we first find a₁₅:

a₁₅ = 4 + (15-1)3 = 4 + 42 = 46

S₁₅ = 15/2 (4 + 46) = 15/2 (50) = 375

Understanding Geometric Sequences

A geometric sequence (or geometric progression) is a sequence where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted by 'r'.

Key Characteristics:

Constant Ratio: The defining characteristic is the consistent ratio between consecutive terms.
Exponential Growth: The terms grow (or decrease) exponentially. Graphing a geometric sequence results in an exponential curve.
Formula: The nth term of a geometric sequence can be found using the formula: aₙ = a₁ * r⁽ⁿ⁻¹⁾ where:
- aₙ is the nth term
- a₁ is the first term
- n is the term number
- r is the common ratio

Example 5: Identifying a Geometric Sequence

Consider the sequence: 3, 6, 12, 24, 48...

Let's calculate the ratio between consecutive terms:

6 / 3 = 2
12 / 6 = 2
24 / 12 = 2
48 / 24 = 2

The common ratio is 2. Therefore, this is a geometric sequence.

Example 6: Finding the nth term

Let's find the 8th term (a₈) of the sequence from Example 5, where a₁ = 3 and r = 2.

Using the formula: aₙ = a₁ * r⁽ⁿ⁻¹⁾

a₈ = 3 * 2⁽⁸⁻¹⁾ = 3 * 2⁷ = 3 * 128 = 384

The 8th term is 384.

Example 7: Finding the common ratio

Given that the 2nd term (a₂) of a geometric sequence is 10 and the 5th term (a₅) is 80, find the common ratio.

We can set up two equations using the formula:

a₂ = a₁ * r¹ = 10 a₅ = a₁ * r⁴ = 80

Dividing the second equation by the first:

(a₁ * r⁴) / (a₁ * r) = 80 / 10 r³ = 8 r = 2

The common ratio is 2.

Sum of a Geometric Sequence

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

Sₙ = a₁ (1 - rⁿ) / (1 - r) where r ≠ 1

Example 8: Finding the sum

Find the sum of the first 6 terms of the geometric sequence 2, 6, 18, 54...

Here, a₁ = 2, r = 3, and n = 6. Using the formula:

S₆ = 2 (1 - 3⁶) / (1 - 3) = 2 (1 - 729) / (-2) = 728

Therefore, the sum of the first 6 terms is 728.

Infinite Geometric Series

When the absolute value of the common ratio |r| is less than 1, the geometric series converges to a finite sum, even if it has infinitely many terms. The sum of an infinite geometric series is given by:

S = a₁ / (1 - r) where |r| < 1

Example 9: Sum of an Infinite Geometric Series

Find the sum of the infinite geometric series 1, 1/2, 1/4, 1/8...

Here, a₁ = 1 and r = 1/2. Since |r| = 1/2 < 1, the series converges:

S = 1 / (1 - 1/2) = 1 / (1/2) = 2

The sum of the infinite series is 2.

Applications of Arithmetic and Geometric Sequences

Arithmetic and geometric sequences appear in various real-world scenarios:

Simple Interest: The yearly balance in a simple interest account forms an arithmetic sequence.
Compound Interest: The yearly balance in a compound interest account forms a geometric sequence.
Depreciation: The value of an asset depreciating at a constant rate each year follows a geometric sequence.
Population Growth (under certain assumptions): Population growth can be modeled using geometric sequences.
Physics: Many physics problems involve concepts related to arithmetic and geometric sequences.

Understanding these sequences provides a powerful tool for modeling and analyzing various phenomena.

Hw 7.1.1-3 Arithmetic And Geometric Sequences

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