Find the natural number ' a ' for which ∑k=1^n f(a+k)=16(2^n-1), where f satisfies the relation

Find the natural number ' a ' for which ∑k=1^n...

Key Concepts

Functional Equations

Functional equations involve identifying functions that satisfy particular algebraic relations for all inputs in their domains. In this context, the equation f(x+y) = f(x) · f(y) suggests that the function has a multiplicative property over addition, a characteristic commonly associated with exponential functions.

Exponential Functions

Exponential functions are mathematical functions in which a constant base is raised to a variable exponent. Given the functional equation f(x+y) = f(x) · f(y) and the initial condition f(1) = 2, the function is determined to be of an exponential form, specifically f(n) = 2^n when n is a natural number.

Geometric Series Summation

A geometric series is the sum of terms in which each term is a constant multiple (the common ratio) of the preceding term. In the problem, summing f(a+k) results in a geometric series of the form 2^(a+k). The sum of such a series is evaluated using the standard formula for the sum of a finite geometric sequence.

Utilizing Initial Conditions

The initial condition f(1) = 2 plays a crucial role in uniquely determining the function corresponding to the functional equation. It ensures that the function value is fixed and leads to the identification of the exponential function, thereby enabling the matching of the given series sum to deduce the unknown parameter.

Transcript

00:01 Hello everyone from the question we are given that summation k equals to 1 to n f a plus k is equal to 16 into 2 to the power n minus 1 where f satisfy the relation f x plus y is equal to f x into f x for all x belongs to n and f 1 is equal to 2 so we have to find the value of a so for this first we put x equals to 1 and y equals to 1 then we get f1 plus 1 is equal to f1 multiplied by f1.

00:35 This implies f2 is equal to 2 into 2 because f1 equals to 2.

00:41 This implies f2 is equal to 2 to the power 2.

00:45 Now if we put x equals to 2 and y equals to 1, then we get f2 plus 1 is equal to f2 multiplied by f1.

00:55 This is equals to 2 to the power 2 multiply by 2.

00:58 This implies f3 is equal to 2 to the power 3.

01:02 Now if we put x equals to 2 and y equals to 2, then we get f2 plus 2 is equal to f2 multiplied by f2.

01:12 This is equals to 2 to the power 2 multiply by 2 to the power 2.

01:15 This implies f4 is equal to 2 to the power 4.

01:19 Now summation k equals to 1 to n fa plus k.

01:24 This is equals to f a plus 1.

01:27 If we put the value of k equals to 1 to n, then we get f a plus 1 plus fa plus 1 plus fa plus 2 plus fa plus f2 plus fa plus f2 plus fa plus f2 plus fa multiply by fn.

01:51 Because we are given that fx plus y is equals to fx into f1.

01:56 So we can write f a plus 1 equals to f a into f1.

01:59 This is equals to fa multiplied by f1 plus f2 plus f3 plus up to fn.

02:07 This is equals to fa multiply by 2 plus 2 to 2 to the power 2 plus 2 to the power 3 plus up to 2 to the power n...

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