Algebra 2 Problems of the Day (open ended)

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August 2017, Part II

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


29. While experimenting with her calculator, Candy creates the sequence 4, 9, 19, 39, 79, … .
Write a recursive formula for Candy’s sequence.
Determine the eighth term in Candy’s sequence.

Answer:
If you look at the differences, you see that the sequence increases by 5, then 10, then 20, then 40, etc. It does not have a common difference, but it does have a common ratio of 2, so it's a geometric sequence.
The initial value is 4, and 4 times 2 is 8, so you have to add another 1 to get 9, the second number in the sequence.

The recursive formula for this sequence will be:

a₁ = 4
a_n = a_n-1 + 1

To get the 8th term in the sequence, just continue where the question left off. You were given the first 5 terms.
a₆ = a₅ + 1 = 2(79) + 1 = 159
a₇ = a₆ + 1 = 2(159) + 1 = 319
a₈ = a₇ + 1 = 2(319) + 1 = 639

30. In New York State, the minimum wage has grown exponentially. In 1966, the minimum wage was $1.25 an hour and in 2015, it was $8.75. Algebraically determine the rate of growth to the nearest percent.

Answer:
2015 - 1966 = 49 years, which will be the exponent.
8.75 = 1.25(x)⁴⁹
7 = x⁴⁹
(7)^{(1/49) = x --> Yes, you are going to take the 49th root of 7.
x = 1.04051.., which is approximately 104%
This is about 4% growth to the nearest percent.}

(Don't stop at 1.04, or round that to 1)



Comments and questions welcome.

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