Problem 171 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the natural number \(a\) for which \(\sum_{k=1}^{n} f(a+k)=16\left(2^{n}-1\right)\), where the function \(f\) satisfies the relation \(f(x+y)=f(x) \cdot f(y)\) for all natural numbers \(x, y\) and further \(f(1)=2 .\)

Step-by-Step Solution

Answer

The value of \(a\) for which \(\sum_{k=1}^{n} f(a+k) = 16\left(2^{n}-1\right)\) for the given function \(f(x)\) is \(a=\boxed{4}\).

Step 1: Determine an expression for \(f(a+k)\)

Using the given function properties \(f(x+y)=f(x) \cdot f(y)\) and \(f(1)=2\), we can find \(f(a+k)\) for any natural number \(k\): For \(k=1\), we have: \[f(a+1)=f(a) \cdot f(1)\] \[f(a+1)=2f(a)\] For \(k=2\), we have: \[f(a+2)=f(a) \cdot f(2)\] Using that \(f(2)=f(1+1)=f(1)\cdot f(1)=4\), we get: \[f(a+2)=4f(a)\] Let's generalize this for any natural number \(k\): \[f(a+k)=2^k f(a)\]

Step 2: Find the sum \(\sum_{k=1}^{n} f(a+k)\)

Using the expression found in Step 1, we'll now find the sum: \[\sum_{k=1}^{n} f(a+k) = \sum_{k=1}^{n} 2^k f(a)\] Since \(f(a)\) is constant with respect to \(k\), we can factor it out: \[ f(a) \sum_{k=1}^{n} 2^k\] Now, we can evaluate the sum: \[\sum_{k=1}^{n} 2^k = 2^1 + 2^2 + \cdots + 2^n = 2(2^n - 1)\] Thus, the sum becomes: \[f(a) \cdot 2(2^n - 1)\]

Step 3: Compare the sum with the given expression and find \(a\)

We are given that the sum is equal to \(16\left(2^{n}-1\right)\): \[f(a) \cdot 2(2^n - 1) = 16\left(2^{n}-1\right)\] Divide both sides by \(2(2^n - 1)\): \[f(a) = 8\] Recall from Step 1 that \(f(a+k)=2^k f(a)\). We can use this expression for \(k=1\) to determine \(f(a+1)\): \[f(a+1)=2f(a)=2\cdot 8=16\] Now, we can use the property \(f(x+y)=f(x) \cdot f(y)\) to find \(a\). Since we have \(f(a+1)=16\) and \(f(1)=2\), we can write: \[f(a) \cdot f(1) = 16\] Substituting \(f(a)=8\) and \(f(1)=2\), we get: \[8 \cdot 2 = 16\] This equation holds, so we have found the value of \(a\) for which the sum is equal to the given expression: \[a = \boxed{4}\]

Key Concepts

Natural Numbers

Natural Numbers are the set of positive integers that begin from 1 and go upwards without any fractional or decimal parts. They are: 1, 2, 3, 4, 5, and so on. These numbers are often used in counting and ordering, making them a fundamental part of mathematics. Notice here that when we talk about natural numbers, zero is typically excluded, though some definitions include it.

• Natural Numbers are infinite. They never stop.
• They don't include negative numbers, fractions, or decimals.
• Arithmetic operations such as addition, subtraction, multiplication, and division (except division by zero) can be performed on these numbers.

Understanding natural numbers helps in laying the groundwork for advanced mathematical concepts such as summations and products.

Summation

Summation is a mathematical operation that represents the addition of a sequence of numbers. It is denoted by the symbol \Sigma (Sigma), which is the Greek letter for 'S' standing for Sum. This operator allows us to compactly write expressions for adding a series of numbers, especially when the numbers follow a predictable pattern.
For example, the summation \(\sum_{k=1}^{n} f(k)\)represents the sum of all values of the function \(f(k)\) from \(k = 1\) to \(k = n\). This concept is useful for working with large series of numbers without having to write them all out.

• Summation accounts for each term in a sequence.
• It's often used to perform calculations over series efficiently.
• The index of summation is a placeholder, typically denoted as \(k\) in \(\sum .\)

In the original problem, summation allows simplifying complex expressions and comparisons.

Exponential Functions

Exponential Functions are mathematical expressions in which variables appear in the exponent. They have the form \(f(x) = a^{x}\), with \(a\) being a constant base. Exponential functions grow rapidly, and are characterized by their constant multiplicative rate of change.
In many problems like the one we have, exponential functions are used to represent phenomena that grow or decay quickly, such as compound interest, population growth, or radioactive decay.

• The base \(a\) is a constant, which affects the rate of growth or decay (\(a > 1\) leads to growth).
• Exponential functions take the form \(f(x+y) = f(x) \, \cdot \, f(y)\). They illustrate how multiplying powers with the same base works.
• These functions can be transformed using logarithms for more manageable expressions.

When tackling problems, recognizing exponential rules can make complex expressions more interpretable.

Problem Solving

Problem Solving in mathematics involves understanding problems, devising strategies to tackle them, and executing those strategies effectively. Here, our goal is to find a value of a natural number through comprehension and manipulation of functions and their properties.
Effective problem solving includes:

• Breaking down the problem into smaller, manageable parts.
• Recognizing patterns and using known mathematical properties and structures.
• Using logical reasoning and step-by-step methods to reach a solution.

The exercise involves determining a specific natural number \(a\) by applying these structured steps:
\(f(x+y)=f(x)\,\cdot\,f(y)\), a property of exponential functions, and simplifying given equations using known sums for easier computation.
Ultimately, practicing problem-solving builds a strong foundation for tackling various mathematical challenges with confidence.