Problem 35 - College Algebra

Question

In Exercises \(35-36,\) find \(S_{1}\) through \(S_{5}\) and then use the pattern to make a conjecture about \(S_{n}\). Prove the conjectured formula for \(S_{n}\) by mathematical induction. $$ S_{n}: \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+\dots+\frac{1}{2 n(n+1)} $$

Step-by-Step Solution

Answer

The conjectured formula for the given series \(S_{n} = \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+...\) is \(\frac{n}{2(n+1)}\), and this has been proven using the principle of mathematical induction.

Step 1: Calculating Terms S1 to S5

Calculate the given terms \(S_{1}\) through \(S_{5}\) in the series for \(S_{n} = \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+...\) \n\n \(S_{1} = \frac{1}{4}\) \ \n \(S_{2} = \frac{1}{4} + \frac{1}{12} = \frac{1}{3}\) \n\n \(S_{3} = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} = \frac{1}{2}\) \n\n \(S_{4} = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} + \frac{1}{40} = \frac{2}{3}\) \n\n \(S_{5} = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{60} = \frac{5}{6}\)

Step 2: Finding the Pattern

Analyze the results from step 1 to find a pattern.\n\nObserving the pattern from results of \(S_1\) to \(S_5\), the pattern seems to be \(\frac{n}{2n+2}\) or simplified to \(\frac{n}{2(n+1)}\). Thus the conjectured formula for \(S_n\) is \(\frac{n}{2(n+1)}\)

Step 3: Mathematical Induction - Base Case

For mathematical induction, prove that the pattern holds true for the base case, where \(n = 1\).\n\nSubstitute \(n = 1\) into the conjectured formula \(\frac{n}{2(n+1)}\), it becomes \(\frac{1}{2(1+1)} = \frac{1}{4}\) which is equal to \(S_1 = \frac{1}{4}\). Thus, the base case has been proven.

Step 4: Mathematical Induction - Inductive Step

Assume that the formula \(\frac{n}{2(n+1)}\) holds true for \(n = k\), that is \(S_k = \frac{k}{2(k+1)}\)\n\nThen try to prove it holds for \(n = k+1\). So, we expect, \n\n\(S_{k+1} = S_k + \frac{1}{2(k+1)(k+2)}\) \n\nwhen we substitute \(S_k\) from the assumption it becomes .. \n\n\(S_{k+1} = \frac{k}{2(k+1)} + \frac{1}{2(k+1)(k+2)}\) \n\nOn simplification, this becomes \n\n\(S_{k+1} = \frac{k+1}{2((k+1)+1)} \) \n\nThis proves the inductive step of the mathematical induction

Step 5: Conclusion

Having gone through these steps, we can therefore conclude that the formula \(\frac{n}{2(n+1)}\) holds true for all \(n\) in the given series.

Key Concepts

Series Summation

A series is a sequence of numbers that are added together. Series summation is the process of adding up all the terms in a series to find a total, often denoted as \(S_n\) which signifies the sum of the first \(n\) terms of the series.
For the exercise, you calculated \(S_1\) through \(S_5\). Each of these represents the sum of a fraction series up to the nth term:

• \( S_1 = \frac{1}{4} \)
• \( S_2 = \frac{1}{4} + \frac{1}{12} = \frac{1}{3} \)
• \( S_3 = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} = \frac{1}{2} \)
• \( S_4 = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} + \frac{1}{40} = \frac{2}{3} \)
• \( S_5 = \frac{1}{4} + \frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{60} = \frac{5}{6} \)

The numerators and denominators grow in a way dictated by the formula \( \frac{1}{2n(n+1)} \), which forms the fractions added in each term. The series summation is the act of computing these values in sequence, giving insight into how these individual fractions collectively accumulate.

Conjecture

A conjecture is an educated guess or hypothesis formed based on observing patterns or trends. It is used especially in math to predict a general formula or result that can be applied beyond specific cases.
In our exercise, after calculating \(S_1\) through \(S_5\), you found a recurring pattern that led to the conjecture for \(S_n\):

• The formula thought to describe the sum of the series is \( \frac{n}{2(n+1)} \).

This conjecture came from noticing how the calculated sums behave as the number of terms increases. It uses information from known results to extend a prediction about the unknown general term \(S_n\). Proving or disproving such a conjecture is crucial as it allows us to either solidify our prediction or go back to redefine our pattern recognition.

Pattern Recognition

Pattern recognition is identifying recurring sequences or consistent structures within a set of data. In this exercise, you observed a sequence of sums \(S_1\) to \(S_5\), each resulting from a progressive series of fractional additions. Detecting the consistent pattern in how these sums change enabled the formulation of a conjecture.
Through pattern recognition, you identified that the results seemed to follow the formula \( \frac{n}{2(n+1)} \). Recognizing the common factor or structure led to forming a mathematical induction to prove if this pattern holds true for all terms of the series.

• Firstly, you recognized each sum in terms of fourths, twelfths, etc., simplifying it where possible.
• Next, you observed how these numbers fit into a specific formula.
• Finally, this understanding of pattern facilitated using mathematical induction for validation.

Pattern recognition is a fundamental skill in mathematics that aids in forming conjectures and proving them, helping to solve problems logically and efficiently.