Question

The monthly incomes for the first five months of Yuri's tutoring business are $500, $750, $1125, $1687.50, and $2531.25. Write the explicit rule for the sequence of monthly incomes.

A) The explicit rule for the sequence of monthly incomes is: M_n = $500 * 1.5ⁿ⁻¹, where 'n' represents the month number.
B) The explicit rule for the sequence of monthly incomes is: M_n = $500 + (n * 250), where 'n' represents the month number.
C) The explicit rule for the sequence of monthly incomes is: M_n = $500 * n, where 'n' represents the month number.
D) The explicit rule for the sequence of monthly incomes is: M_n = $1000 - (n * 250), where 'n' represents the month number.

Answer 1

The explicit rule for the sequence of monthly incomes is given by option A) Mₙ = $500* 1.5^(ⁿ⁻¹), where 'n' represents the month number.

Step-by-step explanation:

Option A presents the correct explicit rule for the given sequence of monthly incomes. The expression Mₙ = $500* 1.5^(ⁿ⁻¹) accurately represents the pattern observed in the monthly incomes. Breaking down the formula, we see that each month's income is obtained by multiplying the previous month's income by 1.5. This corresponds to the observed increase in income from one month to the next.

Let's verify this with the given sequence:

- For n=1: M₁ = $500 * 1.5^(1-1) = $500

- For n=2: M₂ = $500 * 1.5^(2-1) = $750

- For n=3: M₃ = $500 * 1.5^(3-1) = $1125

- For n=4: M₄ = $500 * 1.5^(4-1) = $1687.50

- For n=5: M₅ = $500 * 1.5^(5-1) = $2531.25

The calculated values match the given monthly incomes, confirming that option A) provides the correct explicit rule for the sequence. This rule reflects the consistent increase of 1.5 times the previous month's income, accurately modeling the growth pattern in Yuri's tutoring business.