Linear Equations

Step 1: Understand the properties of the function

- satisfies:

\qquad - for all ,

\qquad - for all .

- The functional equation suggests behaves like an exponential function. Let’s explore this property:

\qquad - Set : .

\qquad - Since , we have .

- Now, consider the functional equation again. Functions satisfying with and are often of the form . Let’s assume: where is a constant. Check:

\qquad - , which satisfies the equation.

\qquad - for all , which holds for any real .

- So, is a candidate. We’ll determine using the given conditions. Step 2: Use the given condition

- The sequence is an arithmetic progression (AP).

- Let the first term be , and the common difference be . Then:

\qquad - ,

\qquad - ,

\qquad - .

- Given :
- Substitute :
- Simplify:
- Since , take the natural logarithm:
- Thus: where we used . So:
Step 3: Compute the first sum

- ,

- .

- The sum:
- The series is geometric:

\qquad - First term ( ): ,

\qquad - Common ratio: 2,

\qquad - Number of terms: 50.

- Sum of a geometric series :
- So:
- Given:
- Equate:
- Compute the right-hand side:
- Left-hand side:
- So:
- Solve for :
- This fraction is small, but we’ll use it later. First, let’s compute the sum we need. Step 4: Compute the sum

- Indices from 6 to 30: number of terms = .

- ,

- Sum:
- Geometric series from to :

\qquad - First term ( ): ,

\qquad - Last term ( ): ,

\qquad - Number of terms: 25.

- Sum:
- Total from to :
- First five terms ( to ):
- So:
- Compute:
- Thus:
Step 5: Relate the two sums

- From the given:
- We need:
- Divide the two:
- So:
- Numerator ratio:
- This fraction is approximately:
- We already have .

- Product:
- Notice the pattern in the exponents:
- So:
Final Answer: The value of is .