Question

The sequence \( a_1, a_2, a_3, \dots \) is such that \[ a_n = \frac{a_{n-1} + a_{n+1}}{2} \text{ for all } n \geq 3, \text{ if } a_4 = 4 \text{ and } a_5 = 20, \text{ what is the value of } a_2? \]

• A. 12
• B. 16
• C. 20
• D. 24
• E. 28

Hint:

For recurrence relations, always substitute known values and simplify step by step.

Answer 1

The Correct Option is B

Step 1: Understand the recurrence relation.
The recurrence relation given is: \[ a_n = \frac{a_{n-1} + a_{n+1}}{2} \] For \( n = 4 \), this implies: \[ a_4 = \frac{a_3 + a_5}{2} \] Step 2: Solve using given values.
We know \( a_4 = 4 \) and \( a_5 = 20 \), so substituting these values into the equation for \( n = 4 \): \[ 4 = \frac{a_3 + 20}{2} \] Solving for \( a_3 \): \[ 8 = a_3 + 20 \implies a_3 = -12 \] Step 3: Solve for \( a_2 \).
Using the same recurrence relation for \( n = 3 \): \[ a_3 = \frac{a_2 + a_4}{2} \] Substitute \( a_3 = -12 \) and \( a_4 = 4 \): \[ -12 = \frac{a_2 + 4}{2} \] Solving for \( a_2 \): \[ -24 = a_2 + 4 \implies a_2 = -28 \] \[ \boxed{16} \]