Problem 18 - College Algebra

Question

In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) $$ 1+3+3^{2}+\cdots+3^{n-1}=\frac{3^{n}-1}{2} $$

Step-by-Step Solution

Answer

The equation \(1+3+3^{2}+\cdots+3^{n-1} = \frac{3^{n}-1}{2}\) is true for every positive integer \(n\)

Step 1: Base Case

Check whether the formula is true for \( n = 1 \). Substitute \(n=1\) in the formula, we get \(1 = \frac{3^1 - 1}{2}\) which simplifies to \(1 = 1\). Therefore, the assumption is proved correct for \( n=1 \).

Step 2: Inductive Hypothesis

Assume the statement is true for some \( n = k \), or \(1+3+3^{2}+ \cdots +3^{k-1} = \frac{3^k -1}{2}\). This is the inductive hypothesis.

Step 3: Inductive Step

Prove that the statement is true for \( n=k+1 \). Considering \(1+3+3^{2}+ \cdots +3^{k-1}+ 3^k\), adding \(3^{k}\) to both sides of our inductive hypothesis gives \(1+3+3^{2}+ \cdots +3^{k-1} + 3^k= \frac{3^k -1}{2} + 3^k\). Simplifying the right side gives \(\frac{3^{k+1} -1}{2}\), which shows that the statement is true for \( n=k+1 \) when it is true for \( n=k \). This completes the proof.

Key Concepts

Proof by Induction

Understanding mathematical induction is essential for students delving into advanced mathematical concepts. It's a technique often used to prove that a given statement is true for all natural numbers. To carry out a proof by induction, you must perform two crucial steps.

The first step is the base case. You verify the statement for the initial value, usually for the positive integer 1. If the statement holds for the base case, you proceed with the second step, the inductive hypothesis. Here, you assume the statement is true for some arbitrary positive integer 'k'.

Once the inductive hypothesis is set, you embark on the inductive step. This involves proving the statement for 'k+1', typically by building on the assumption that it's true for 'k'. If successful, it confirms the statement holds for every number that follows the base case, symbolically sealing the truth of the statement for all natural numbers. The process, akin to a domino effect, secures a logical foundation for innumerable mathematical assertions.

Sequence and Series

Sequence and series form the bedrock of many mathematical patterns, encapsulating an ordered list of numbers where each number is called a term. A sequence is just that list, while a series is the sum of the terms of the sequence.

Understanding the difference between these two is crucial. A sequence is simply an ordered set of numbers, and when we talk about a series, we're focusing on the sum of the sequence's elements. This concept applies not just in pure mathematics but pervades other fields like physics, economics, and various forms of engineering.

In our exercise, we are dealing with a finite portion of a series, which can be summed up to provide a fixed number. This sum is particularly important for determining patterns or behaviors within a set of numbers, which often has practical applications in the real world like computing interest or forecasting trends.

Geometric Series

A geometric series is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the exercise provided, the common ratio is 3, making the series geometric.

One of the remarkable features of geometric sequences is that they have a special formula to calculate the sum of their first 'n' terms. This significantly simplifies the process of solving problems where large sequences are involved. In the case of our exercise, the sum of the first 'n' terms of the series \(1 + 3 + 3^2 + \cdots + 3^{n-1}\) is represented by the formula \(\frac{3^n - 1}{2}\).

The elegance of a geometric series is best observed through exercises such as these, offering a tangible method to encapsulate potentially infinitely long patterns with a single defining expression. Mastering the use and manipulation of these series can open doors to understanding exponential growth and decay, among other vital mathematical and real-world phenomena.