Problem 46 - Beginning and Intermediate Algebra

Question

Write down the two formulas for \(S_{n},\) and explain when to use each formula.

Step-by-Step Solution

Answer

The two formulas for the sum of the first n terms, \(S_n\), are for arithmetic and geometric progressions: 1. Arithmetic progression: \(S_n = \frac{n(a_1 + a_n)}{2}\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term. Use this formula when the sequence has a constant difference between consecutive terms. 2. Geometric progression: \(S_n = \frac{a_1(r^n - 1)}{r - 1}\) for \(r \neq 1\), and \(S_n = na_1\) for \(r = 1\), where \(n\) is the number of terms, \(a_1\) is the first term, \(r\) is the common ratio between consecutive terms, and \(r^n\) is the ratio after n steps. Use this formula when the sequence has a constant ratio between consecutive terms.

Step 1: 1. Identify the type of sequence

First, determine whether the given sequence is an arithmetic progression or a geometric progression. An arithmetic progression is a sequence where the difference between consecutive terms is constant, while a geometric progression is a sequence where the ratio between the consecutive terms is constant.

Step 2: 2. Write down the formula for the sum of the first n terms in an arithmetic progression

The formula for the sum of the first n terms in an arithmetic progression is \(S_n = \frac{n(a_1 + a_n)}{2}\), where - \(S_n\) represents the sum of the first n terms, - \(n\) represents the number of terms in the sequence, - \(a_1\) represents the first term, and - \(a_n\) represents the nth term in the sequence.

Step 3: 3. Write down the formula for the sum of the first n terms in a geometric progression

The formula for the sum of the first n terms in a geometric progression is \(S_n = \frac{a_1(r^n - 1)}{r - 1}\) for \(r \neq 1\), and \(S_n = na_1\) for \(r = 1\), where - \(S_n\) represents the sum of the first n terms, - \(n\) represents the number of terms in the sequence, - \(a_1\) represents the first term, - \(r\) represents the common ratio between consecutive terms, and - \(r^n\) (raised to the power of n) denotes the ratio between consecutive terms after n steps.

Step 4: 4. Explain when to use each formula

The appropriate formula to use depends on the type of sequence given: - Use the arithmetic progression formula \(S_n = \frac{n(a_1 + a_n)}{2}\) when the sequence exhibits a constant difference between consecutive terms. - Use the geometric progression formula \(S_n = \frac{a_1(r^n - 1)}{r - 1}\) for \(r \neq 1\), and \(S_n = na_1\) for \(r = 1\) when the sequence exhibits a constant ratio between consecutive terms.

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers where each term after the first is generated by adding a constant difference to the previous term.
For example, in the sequence 2, 4, 6, 8, ..., the difference between consecutive terms is 2.
This constant difference is key to identifying an arithmetic progression.

• To find any term in an arithmetic sequence, use the formula: \( a_n = a_1 + (n-1)d \)
• Where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference.

When needing to find the sum of the first n terms, apply the sum formula:\[S_n = \frac{n(a_1 + a_n)}{2}\] - Here, \( S_n \) denotes the sum of the first n terms.- This formula is tailor-made for arithmetic progressions since it relies on both the first and the last term. Understanding and recognizing arithmetic sequences are essential, especially when trying to sum them efficiently.

Geometric Progression

A geometric progression, on the other hand, involves multiplying a term by a constant ratio to get to the next term in the sequence.
For instance, in the sequence 3, 6, 12, 24, ..., the constant multiplied ratio is 2.
This constant ratio characterizes geometric progressions.

• To find any specific term, use: \( a_n = a_1 \cdot r^{(n-1)} \)
• Where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.

For calculating the sum of the first n terms of a geometric progression, use:\[S_n = \frac{a_1(r^n - 1)}{r - 1}, \quad \text{if } r eq 1\]For simple cases where \( r = 1 \):\[S_n = na_1\]- Here, \( S_n \) is the sum, \( a_1 \) is the starting term, and the formula takes into account the growth factor \( r \). These formulas highlight how geometric series function fundamentally, adding up terms that grow or shrink by a consistent factor. They are valuable in scenarios where exponential growth or decay is involved.

Sequence Formulas

Sequence formulas are mathematical expressions that help you find specific terms in a sequence or sum them up. There are key differences, and correct identification of the sequence type is necessary.
Arithmetic sequences grow by addition and follow linear paths, summed up by:

• \( S_n = \frac{n(a_1 + a_n)}{2} \)

Geometric sequences grow or shrink by multiplication and follow exponential paths, with a sum calculated by:

• \( S_n = \frac{a_1(r^n - 1)}{r - 1} \) for \( r eq 1 \)
• \( S_n = na_1 \) for a constant ratio of 1.

These formulas can be applied practically in real-world contexts such as financial calculations, physics, and more.
The accurate application of these formulas ensures consistency and precision when analyzing sequences.