Problem 34 - Algebra and Trigonometry

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. \(-1,1,-1,1, \dots\)

Step-by-Step Solution

Answer

The partial sums are: \(S_1 = -1, S_2 = 0, S_3 = -1, S_4 = 0, S_5 = -1, S_6 = 0\).

Step 1: Identify the Sequence Terms

The given sequence is \(-1, 1, -1, 1, \dots\).These are alternating terms starting with -1.

Step 2: Compute the First Partial Sum \(S_1\)

The first partial sum \(S_1\) is simply the first term of the sequence: \[ S_1 = -1 \]

Step 3: Compute the Second Partial Sum \(S_2\)

Add the first two terms of the sequence to obtain \(S_2\): \[ S_2 = -1 + 1 = 0 \]

Step 4: Compute the Third Partial Sum \(S_3\)

Add the first three terms of the sequence to find \(S_3\): \[ S_3 = -1 + 1 - 1 = -1 \]

Step 5: Compute the Fourth Partial Sum \(S_4\)

Add the first four terms of the sequence to determine \(S_4\): \[ S_4 = -1 + 1 - 1 + 1 = 0 \]

Step 6: Compute the Fifth Partial Sum \(S_5\)

Add the first five terms of the sequence to find \(S_5\): \[ S_5 = -1 + 1 - 1 + 1 - 1 = -1 \]

Step 7: Compute the Sixth Partial Sum \(S_6\)

Add the first six terms of the sequence to obtain \(S_6\): \[ S_6 = -1 + 1 - 1 + 1 - 1 + 1 = 0 \]

Key Concepts

Alternating Sequence

An alternating sequence is a series of numbers where the terms switch between positive and negative. This switching typically follows a consistent pattern of signs. For example, in the sequence - -1, 1, -1, 1, ..., - the signs alternate between negative and positive, beginning with -1. This behavior is purposeful, often aligning with alternating operations or properties in mathematical problems. Alternating sequences are essential because they offer a dynamic way to handle sums and series. These sequences can be helpful in balancing equations or exploring properties in mathematical structures. Whenever you see a sequence that alternates in this manner, take note of the pattern, as it will guide you through the calculations effectively.

Partial Sum Calculation

Partial sums in a sequence are cumulative totals achieved by adding terms from the start of the sequence up to a specified index. Essentially, these sums show how the total accumulation of values from the sequence grows as you include more terms. For instance, if you consider the sequence - -1, 1, -1, 1, ... , - and find the first six partial sums, look at:

• First sum (\( S_1 \)): Just the first term: \(-1\).
• Second sum (\( S_2 \)): Add the first two terms: \(-1 + 1 = 0\).
• Third sum (\( S_3 \)): Add the first three terms: \(-1 + 1 - 1 = -1\).
• Fourth sum (\( S_4 \)): Add the first four terms: \(-1 + 1 - 1 + 1 = 0\).
• Fifth sum (\( S_5 \)): Add the first five terms: \(-1 + 1 - 1 + 1 - 1 = -1\).
• Sixth sum (\( S_6 \)): Add the first six terms: \(-1 + 1 - 1 + 1 - 1 + 1 = 0\).

Partial sum calculations help track cumulative changes and can reveal interesting structures, like periodicity or convergence in infinite sequences. They're a fundamental tool in analyzing mathematical sequences.

Finite Series

A finite series is essentially a sum of a certain number of terms of a sequence. Unlike infinite series, a finite series has a definite endpoint. In the case of our sequence - -1, 1, -1, 1, ..., - this means we are only considering the sums up to the sixth term, such as - \( S_6 = 0 \).Finite series are widely used in mathematics and various applied fields since they allow you to handle precise calculations without diving into the complexities of infinite sums. They help understand the partial behaviors and patterns of sums,- - revealing regularities- or specific characteristics within simplified bounds.Understanding how to manage and sum a finite series is crucial, especially when dealing with alternating sequences. This segmentation into finite portions makes it simpler to handle, explore, and apply in practical scenarios.