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# Algebra/Recursive Sequence

Question
You complain that your hotel room is too cold. The hotel tells you that they will increase the temperature by 8% each hour. If the current temperature of the room is 72 degrees F, what will the temperature be after 4 hours, to the nearest tenth of a degree?

Is this sequence geometric or arithmetic?

Write a recursive formula for the sequence.

Let t(n) be the temperature in the room after n hours.

The room starts at 72 , so

t(0) = 72

After 1 hour, the temperature increases by 8%

t(1) = (1.08)(72) = 77.76

After 2 hours , the temperature increases by 8% again

t(2) = (1.08)(77.76) = 83.98

continue in this way , multiplying by 1.08 two more times

t(3) = (1.08)(83.98) = 90.7

t(4) = (1.08)(90.7) = 97.95

After 4 hours , the temperature will be 98 degrees , to the nearest tenth.

Since we keep finding the next temperature in the sequence by multiplying by 1.08 , the sequence is geometric.

A recursive formula for the sequence is t(n+1) = (1.08)t(n)  , n = 0,1,2....,  with t(0) = 72

This gives t(n) = (72)(1.08)^n

Algebra

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#### Socrates

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