Question

write an equation that models the sequence 200, 500, 1250, ...
a y = 120x + 80
b y = 80(0.4)^x
c y = 80 (2.5)^x
d y = 80 (120)^x

Explanation:

Step1: Identify Sequence Type

Check if the sequence is geometric (common ratio) or arithmetic (common difference).

• Arithmetic: \(500 - 200 = 300\), \(1250 - 500 = 750\) (not constant, so not arithmetic).
• Geometric: \(\frac{500}{200} = 2.5\), \(\frac{1250}{500} = 2.5\) (common ratio \(r = 2.5\)).

Step2: Recall Geometric Sequence Formula

The general form of a geometric sequence is \(y = a(r)^x\), where \(a\) is the initial term, \(r\) is the common ratio.

Step3: Determine Initial Term \(a\)

Test \(x = 1\) (first term, 200) in the options:

• Option C: \(y = 80(2.5)^1 = 80 \times 2.5 = 200\) (matches the first term).
• Check \(x = 2\): \(80(2.5)^2 = 80 \times 6.25 = 500\) (matches second term).
• Check \(x = 3\): \(80(2.5)^3 = 80 \times 15.625 = 1250\) (matches third term).

Answer:

C. \(y = 80(2.5)^x\)