Question

An employee joined a company in 2020 with a starting salary of $60,000. Every year, the employee receives a raise of $1,000 plus 3% of the salary of the previous year.

(a). Set up a recurrence relation for the salary of the employee n years after 2020. A recurrence relation of a sequence is an equation that expresses aₙ using one or more of previous terms.
(b). What will be the salary of the employee in 2025, 5 years after the start of working?
(c). Find an explicit formula for the salary n years after 2020.

Answer 1

The recurrence relation for the salary of the employee can be expressed as aₙ = aₙ₋₁ + 1,000 + 0.03 * aₙ₋₁. The salary of the employee in 2025 will be $69,570.11. An explicit formula for the salary n years after 2020 is aₙ = $60,000 + $1,000 * n + $30 * n(n-1).

Step-by-step explanation:

(a). The recurrence relation for the salary of the employee n years after 2020 can be expressed as:

an = an-1 + 1,000 + 0.03 * an-1

(b). To find the salary of the employee in 2025 (5 years after the start of working), we can use the recurrence relation to calculate it step by step. Starting with a0 = $60,000, we have:

1. a1 = a0 + 1,000 + 0.03 * a0 = $61,800
2. a2 = a1 + 1,000 + 0.03 * a1 = $63,654
3. a3 = a2 + 1,000 + 0.03 * a2 = $65,565.62
4. a4 = a3 + 1,000 + 0.03 * a3 = $67,536.98
5. a5 = a4 + 1,000 + 0.03 * a4 = $69,570.11

Therefore, the salary of the employee in 2025 will be $69,570.11.

(c). An explicit formula for the salary n years after 2020 can be derived by solving the recurrence relation. By simplifying the equation, we get:

an = $60,000 + $1,000 * n + $30 * n(n-1)