Understanding the Function Property f(x+y)=f(x)f(y)

The given condition \(f(x + y) = f(x) f(y)\) is a well-known functional equation. Functions that satisfy this property for all real or natural numbers \(x, y\) are typically exponential functions of the form \(f(x) = a^x\) for some constant \(a\).

Determining the Function f(x)

We are given that the function \(f(x)\) satisfies \(f(x + y) = f(x) f(y)\) for all \(x, y \in \mathbb{N}\). We are also given that \(f(1) = 2\).

Let's use the property to find the function form. Assuming \(f(x) = a^x\):

• \(f(x+y) = a^{x+y}\)
• \(f(x)f(y) = a^x \cdot a^y = a^{x+y}\)

This confirms that \(f(x) = a^x\) is a valid form for this property.

Now, using the condition \(f(1) = 2\):

\(f(1) = a^1 = a\)

Since \(f(1) = 2\), we have \(a = 2\).

Therefore, the function is \(f(x) = 2^x\).

Calculating the Summation \(\displaystyle\sum_{x=1}^5\) f(2x − 1)

We need to evaluate the sum \(\displaystyle\sum_{x=1}^5\) \(f(2x - 1)\). We have found that \(f(x) = 2^x\). So, \(f(2x - 1) = 2^{2x - 1}\).

The summation becomes \(\displaystyle\sum_{x=1}^5\) \(2^{2x - 1}\).

Let's write out the terms of the sum for \(x = 1, 2, 3, 4, 5\):

• For \(x = 1\): \(f(2(1) - 1) = f(1) = 2^{2(1)-1} = 2^1 = 2\)
• For \(x = 2\): \(f(2(2) - 1) = f(3) = 2^{2(2)-1} = 2^3 = 8\)
• For \(x = 3\): \(f(2(3) - 1) = f(5) = 2^{2(3)-1} = 2^5 = 32\)
• For \(x = 4\): \(f(2(4) - 1) = f(7) = 2^{2(4)-1} = 2^7 = 128\)
• For \(x = 5\): \(f(2(5) - 1) = f(9) = 2^{2(5)-1} = 2^9 = 512\)

The sum is \(2 + 8 + 32 + 128 + 512\).

Identifying the Geometric Series

The terms of the sum are \(2, 8, 32, 128, 512\). Let's check the ratio between consecutive terms:

• \(8 / 2 = 4\)
• \(32 / 8 = 4\)
• \(128 / 32 = 4\)
• \(512 / 128 = 4\)

Since the ratio between consecutive terms is constant, this is a geometric series with:

• First term, \(a = 2\)
• Common ratio, \(r = 4\)
• Number of terms, \(n = 5\)

Summing the Geometric Series

The sum \(S_n\) of a geometric series with first term \(a\), common ratio \(r\), and \(n\) terms is given by the formula:

\(S_n = \frac{a(r^n - 1)}{r - 1}\)

For our series, \(a=2\), \(r=4\), and \(n=5\):

\(S_5 = \frac{2(4^5 - 1)}{4 - 1}\)

Calculate \(4^5\):

\(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\)

Now substitute this value back into the sum formula:

\(S_5 = \frac{2(1024 - 1)}{3}\)

\(S_5 = \frac{2(1023)}{3}\)

Divide 1023 by 3:

\(1023 \div 3 = 341\)

Now multiply by 2:

\(S_5 = 2 \times 341 = 682\)

Final Summation Result

The value of \(\displaystyle\sum_{x=1}^5\) \(f(2x - 1)\) is 682.

Revision Table: Key Steps to Calculate the Summation

Additional Information: Functional Equations and Series

Functional equations are equations where the unknowns are functions. The equation \(f(x+y) = f(x)f(y)\) is a classic example, known as Cauchy's exponential functional equation. Its continuous solutions on real numbers are of the form \(f(x) = a^x\). For solutions defined only on natural numbers or integers, the same form \(f(x) = a^x\) holds, where \(a = f(1)\).

A geometric series is a series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of the first \(n\) terms of a geometric series can be calculated using a specific formula, which is very useful in many mathematical and real-world applications.