Problem 18 - College Algebra

Question

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1+3+3^{2}+\dots+3^{n-1}=\frac{3^{n}-1}{2}$$

Step-by-Step Solution

Answer

By the principle of mathematical induction, we have proven that for any positive integer \(n\), the sum of the geometric series \(1+3+3^{2}+\dots+3^{n-1}\) is given by \(\frac{3^{n}-1}{2}\).

Step 1: Proof Base Case

First, verify the base case. That is, check if the statement is true for \(n = 1\). The left hand side (LHS) of the equation with \(n = 1\) is simply 1, while the right hand side (RHS) of the equation for \(n = 1\) is also 1. Hence, the equality holds for \(n = 1\).

Step 2: Proof Inductive Hypothesis

Next, make the inductive hypothesis. Assume that the statement is true for some positive integer \(n = k\). That is, \(1+3+3^{2}+\dots+3^{k-1}=\frac{3^{k}-1}{2}\). This is assumed to hold true.

Step 3: Inductive Step

Now, we proceed with the inductive step to prove that if the statement is true for \(n = k\), it must also be true for \(n = k +1\). Consider the LHS of the equation for \(n = k+1\). It can be expressed as \(1+3+3^{2}+\dots+3^{k-1}+3^k\). By the inductive hypothesis, the sum of the first \(k\) terms is \(\frac{3^{k}-1}{2}\). Hence, the LHS can be rewritten as \(\frac{3^{k}-1}{2} + 3^k = \frac{3^{k}-1+2*3^{k}}{2} = \frac{3^{k+1}-1}{2}\). Hence, the equation holds for \(n = k +1\). Therefore, by mathematical induction, the equation holds for all positive integers \(n\).

Step 4: Conclusion

Thus, it has been proven through mathematical induction that for any positive integer \(n\), the sum of this geometric series is as given in the equation.

Key Concepts

Proof by Induction

Proof by induction is a powerful method used to prove that a statement holds true for all positive integers. It's built on a simple yet profound principle: if you can demonstrate that a statement works for an initial case (often starting with 1), and then show that if it's assumed to be true for any integer, it must also be true for the next integer, the statement must be correct for all integers in sequence.

Here's how induction works in three stages: The 'Base Case' tests the validity of the statement for the initial integer, usually when n=1. 'Inductive Hypothesis' is the assumption that the statement is true for some positive integer k. Finally, the 'Inductive Step' links the hypothesis to the case for k+1, completing the logical bridge. If this step is successful, the proof by induction concludes that the statement holds for all positive integers.

Geometric Series

A geometric series is a sequence 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 series 1, 3, 9, 27, ... the common ratio is 3, as each term is three times the one before it.

The sum of a finite geometric series can be found using the formula: \[ S_n = \frac{a(1-r^n)}{1-r} \] where \(S_n\) is the sum of the first n terms, \(a\) is the first term, and \(r\) is the common ratio. The infinite series has a sum only if \(|r| < 1\), and it's given by \(S = \frac{a}{1-r}\). The exercise uses this notion to establish a relationship between the sum of the series and its nth term.

Algebraic Proof

Algebraic proof employs algebra to demonstrate the truthfulness of mathematical statements. It is built upon definitions, theorems, properties, and logical reasoning to establish the validity of a statement beyond any doubt. When dealing with geometric series, algebraic manipulation is critical in simplifying the terms and formulating an equation that proves the sum.

As shown in our exercise, transforming the left-hand side of the equation using algebraic identities and the inductive hypothesis allows the proof to flow and show that a property holds true. This clarity is vital in moving from one step to the next, using arithmetic operations and rearrangements to derive the ultimate result that supports the statement.

Positive Integers

Positive integers are the set of numbers 1, 2, 3, and so on, which represent quantities and can be used to count objects. They are foundational in understanding mathematical induction since induction proofs typically run over the domain of positive integers. When we talk about sequences or series such as geometric series, positive integers typically index the terms of the series. In our exercise, the sum of the series is expressed for every positive integer value of n. The mathematical induction method proves this summation formula for all positive integers, solidifying its universal truth within this domain.