Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Sequence of Partial Sums In Exercises 51 and \(52,\) find the sequence of the first five partial sums \(S_{1}, S_{2}\) \(S_{3}, S_{4},\) and \(S_{5}\) of the geometric sequence by adding terms. $$8,-4,2,-1, \frac{1}{2}, \ldots$$

Step-by-Step Solution

Answer

The sequence of the first five partial sums is: \(S_1 = 8\), \(S_2 = 4\), \(S_3 = 6\), \(S_4 = 5\), and \(S_5 = 5.5\).

Step 1: Identify the first term

The first term of the sequence, \(a_1\), is 8.

Step 2: Calculate the first partial sum \(S_1\)

The first partial sum \(S_1\) equals the first term of the sequence, which is 8.

Step 3: Calculate the second partial sum \(S_2\)

The second partial sum \(S_2\) is found by adding the first and the second term, \(8 + (-4) = 4\).

Step 4: Calculate the third partial sum \(S_3\)

The third partial sum \(S_3\) is found by adding the previous total and the third term, \(4 + 2 = 6\).

Step 5: Calculate the fourth partial sum \(S_4\)

The fourth partial sum \(S_4\) is found by adding the previous total and the fourth term, \(6 + (-1) = 5\).

Step 6: Calculate the fifth partial sum \(S_5\)

The fifth partial sum \(S_5\) is found by adding the previous total and the fifth term, \(5 + \frac{1}{2} = 5.5\).

Key Concepts

Geometric Sequences

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the sequence \(8, -4, 2, -1, \frac{1}{2}, \ldots\), the common ratio is \(-\frac{1}{2}\). Every number is \(-\frac{1}{2}\) times the number before it. This pattern continues indefinitely unless a sequence has a fixed number of terms.

Geometric sequences can be a broadly used mathematical concept and appear in various fields, including physics, finance, and computer science, leading to a deeper understanding of exponential growth and decay patterns.

Series and Summation

When we talk about series in mathematics, we refer to the sum of the terms of a sequence. Specifically, a geometric series is the sum of the terms of a geometric sequence. The process of adding these terms together is known as summation. Summation notation can be used to represent the series compactly, using the Greek letter sigma, \(\Sigma\), for the sum. For instance, the sequence of partial sums \(S_1, S_2, S_3, S_4,\) and \(S_5\) from our example sequence involves adding together the first one, two, three, and so on terms of the sequence to find each respective partial sum.

This process is fundamental in understanding infinite series and evaluating their convergence to determine if the sum approaches a finite limit.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators, variables, and grouping symbols. When solving for the partial sums of a geometric sequence, we use algebraic expressions to represent the sums at each step.

For example, the second partial sum (\(S_2\)) in our given sequence is an algebraic expression \(8 + (-4)\), which simplifies to \(4\). Each partial sum is an algebraic expression that represents the accumulated total of the sequence's terms up to a certain point.

Sequence Convergence

Sequence convergence is a concept where we determine if a sequence of numbers approaches a specific value as the number of terms increases indefinitely. In the context of geometric sequences, a series converges if the absolute value of the common ratio is less than \(1\). This is because each successive term will get smaller and infinitely close to zero. However, if the common ratio's absolute value is \(1\) or greater, the series does not converge and instead increases or decreases without bound.

When discussing the partial sums of a geometric sequence, understanding convergence helps us recognize whether the infinite sum of the sequence will have a finite limit or whether it will diverge to infinity.