Question

Let \( \langle a_n \rangle \) be a sequence such that \( a_0 = 0 \), \( a_1 = \frac{1}{2} \), and \( 2a_{n+2} = 5a_{n+1} - 3a_n \).n= 0,1,2,3.... Then \( \sum_{k=1}^{100} a_k \) is equal to:

• A. \( 3a_{99} + 100 \)
• B. \( 3a_{99} - 100 \)
• C. \( 3a_{100} + 100 \)
• D. \( 3a_{100} - 100 \)

Hint:

Utilize characteristic equations to solve linear recurrence relations efficiently.

Answer 1

The Correct Option is C

Step 1: Solve the recurrence relation.
Solve for \( a_{100} \) using methods suitable for linear homogeneous recurrence relations.
Step 2: Compute the sum.
Sum up the sequence based on the relationship \( \sum a_k = 3a_{100} + 100 \).
Conclusion: Thus, \( \sum_{k=1}^{100} a_k = 3a_{100} + 100 \).