Problem 30 - Algebra and Trigonometry

Question

Use mathematical induction to prove that each statement is true for every positive integer n. \(\sum_{i=1}^{n} 7 \cdot 8^{i}=8\left(8^{n}-1\right)\)

Step-by-Step Solution

Answer

The assertion is proven true by mathematical induction. It holds for the base case (\(n=1\)). Assuming it's true for \(n=k\), it's proven true for \(n=k+1\). Therefore, it holds for all positive integers \(n\).

Step 1: Verification of base case

The base case here is for \(n=1\). Let's calculate the left and right side when \(n=1\). Left hand side(LHS)=\(\sum_{i=1}^{1} 7 \cdot 8^{i}= 7 \cdot 8 =56\). Right hand side(RHS)=\(8\left(8^{1}-1\right)=8(8-1)=8*7=56\). So, LHS=RHS which verifies the formula for the base case.

Step 2: Inductive Step: Assumption

Next, let's make the inductive assumption. Assume that the formula holds for some positive integer \(k\), i.e., \(\sum_{i=1}^{k} 7 \cdot 8^{i}=8\left(8^{k}-1\right)\). This will be used as the inductive hypothesis.

Step 3: Inductive Step: Verification

Next, we will use the hypothesis to prove the formula for \(n=k+1\). Let's consider the left-hand side of the formula for \(n=k+1\), \(\sum_{i=1}^{k+1} 7 \cdot 8^{i}\). It can be rewritten as \(\sum_{i=1}^{k} 7 \cdot 8^{i}+ 7 \cdot 8^{k+1}\). From the hypothesis, we know \(\sum_{i=1}^{k} 7 \cdot 8^{i}=8\left(8^{k}-1\right)\), so we can substitute this, resulting in \(8\left(8^{k}-1\right)+ 7 \cdot 8^{k+1} = 8^{k+1}+ 7 \cdot 8^{k+1} - 8 = 8(8^{k+1}-1)\). So, the original formula holds for \(n=k+1\) and therewith it is proven by induction.

Key Concepts

Proof by Induction

Mathematical induction is a powerful technique used to prove statements, especially those involving sequences or sums. Imagine it as a chain of dominoes; if one falls, and each subsequent is close enough to knock the next over, they'll all fall. Induction leverages the same principle.The process has two stages:

• Base case: Show that the statement is true for the first instance (often when \( n = 1 \)).
• Inductive step: Assume the statement holds for a number \( k \), and then show it's also true for \( k + 1 \).

This structured approach leads us to prove that a statement works for all positive integers.

Positive Integers

Positive integers are simply the set of numbers \( 1, 2, 3, \) and so on. They are the building blocks for many mathematical concepts and are pivotal in induction proofs.When proving statements concerning positive integers, induction provides a natural framework. It works seamlessly with these numbers due to their ordered nature and the absence of negative values or zero.In our current example, the induction process starts from the smallest positive integer \( n = 1 \) and extends its logic beyond, showing how it holds indefinitely as you move through the numbers.

Summation Formula

A summation formula is a concise way to describe the addition of a sequence of numbers. Our example \[ \sum_{i=1}^{n} 7 \cdot 8^{i} = 8(8^{n} - 1) \]illustrates a pattern where each term is a multiple of 8, which grows exponentially.Understanding the summation is crucial when applying induction. In the induction process, you use a summation formula to make an assumption and later verify how this assumption lets you prove further cases. The formula is both a starting point and a tool within the induction framework.

Base Case and Inductive Step

The base case and inductive step are the two essential components of any proof via induction.

Base Case

Start with the simplest scenario, usually \( n = 1 \). We showed:- LHS: \( \sum_{i=1}^{1} 7 \cdot 8^{i} = 56 \)- RHS: \( 8(8 - 1) = 56 \)Since LHS equals RHS, the base case holds.

Inductive Step

Assume the formula is true for some arbitrary positive integer \( k \). Here, \[ \sum_{i=1}^{k} 7 \cdot 8^{i} = 8(8^{k} - 1) \]Next, use this to show it's true for \( k+1 \):- LHS for \( n = k+1 \) becomes: \( \sum_{i=1}^{k} 7 \cdot 8^{i} + 7 \cdot 8^{k+1} \)- Substitute: \( 8(8^{k} - 1) + 7 \cdot 8^{k+1} \)- Simplify to: \( 8(8^{k+1} - 1) \)Completing the induction step proves that the formula is universally true for all positive integers.