Problem 163 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that \(\sum_{r=1}^{n} \log \left(\frac{a^{r}}{b^{r-1}}\right)=\frac{n}{2} \log \left(\frac{a^{n+1}}{b^{n-1}}\right)\)

Step-by-Step Solution

Answer

The proof using mathematical induction shows that the given equation is true for n = 1 (base case) and if it holds for n, it is true for n+1 as well (inductive step). Therefore, it can be concluded that the given equation is true for all positive integers n.

Step 1: Base Case

The first thing that needs to be done is to test the base case, i.e. n = 1. Plugging it into both sides of the equation, we get, \(\sum_{r=1}^{1} \log \left(\frac{a^{r}}{b^{r-1}}\right) = \frac{1}{2} \log \left(\frac{a^{2}}{b^{0}}\right)\). Simplifying both sides, we find they are equal, thus the base case holds.

Step 2: Inductive Step: Assumption

Assume the equation holds for n, i.e. \(\sum_{r=1}^{n} \log \left(\frac{a^{r}}{b^{r-1}}\right) = \frac{n}{2} \log \left(\frac{a^{n+1}}{b^{n-1}}\right)\). This is known as inductive hypothesis.

Step 3: Inductive Step: Inductive Conclusion

Now it needs to be shown that the equation holds for n+1. So adding the term for n+1 to both sides of the inductive hypothesis and simplifying, we get, \[\frac{n+1}{2} \log \left(\frac{a^{n+2}}{b^{n}}\right) = \sum_{r=1}^{n+1} \log \left(\frac{a^{r}}{b^{r-1}}\right)\]This proves the equation holds for n+1 when it holds for n.

Key Concepts

Logarithms

Logarithms are fundamental components of algebra that help solve equations involving exponential functions. A logarithm, typically written as \( \log_b a \), asks the question 'To what power must we raise the base \( b \) to obtain \( a \)?' To grasp this during problem-solving, one needs to be familiar with the logarithm laws, such as the product, quotient, and power rules.

In the given exercise, we see a sequence involving the logarithm of a quotient. Simplifying such expressions requires using the properties of logarithms. The quotient rule, \( \log_b \frac{a}{c} = \log_b a - \log_b c \), is particularly useful here as it allows the separation of the logarithm of a quotient into the difference of two logarithms, which simplifies the equation at hand.

Understanding logarithms and their properties is essential to manipulating and simplifying sequences like those in the exercise, which ensures a smoother application of mathematical induction techniques.

Sequences and Series

Sequences and series are integral parts of mathematics that deal with ordered lists of numbers and their summations, respectively. A sequence is a set of numbers in a specific order, while a series is the sum of a sequence of terms.

The problem presented revolves around a particular type of series where each term involves a logarithmic function. The summation notation \( \sum_{r=1}^{n} \) indicates the series with 'r' being the variable that denotes the term number, and 'n' is the last term number in the sequence.

Series can often be evaluated by recognizing patterns or by using formulas derived from such patterns. The exercise showcases how sequences and logarithms intersect, urging students to understand both concepts to solve complex logarithmic sequences. Recognizing the role of each component within the series is crucial for proper application of mathematical induction.

Proofs are the essence of mathematics and are used to establish the truth of a statement beyond doubt. Mathematical induction, a common proof technique, consists of two main steps: verifying the base case and proving that if the statement holds for \( n \), it must also hold for \( n+1 \).

In the given exercise, the base case is checked first to confirm that the property holds for \( n = 1 \). Then, the inductive step assumes the statement is true for \( n \) and demonstrates it for \( n+1 \) - this is the heart of induction. Inductive proofs are powerful because they show that a statement is true for every possible natural number \( n \), no matter how large.

The concept of induction, particularly when paired with the comprehension of sequences and the properties of logarithms, equips students to prove complex mathematical theorems and facilitates the progression from a specific case to a universal truth. Understanding and mastering proofs provide a strong foundation for all mathematical reasoning.