Question

The recursive formula, an = an-1 - 12 with a0 = 84 describes the amount of water left, in gallons, in a bathtub n after the drain stopper was pulled. Find a5, he amount of water left in the tub after 5 minutes.

Answer 1

Answer: 24 gallons

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Work Shown:

`a_n = a_(n-1) - 12\\\\a_1 = a_(1-1) - 12\\\\a_1 = a_(0) - 12\\\\a_1 = 84 - 12\\\\a_1 = 72\\\\`

After 1 minute, there are 72 gallons of water left.

`a_n = a_(n-1) - 12\\\\a_2 = a_(2-1) - 12\\\\a_2 = a_(1) - 12\\\\a_2 = 72 - 12\\\\a_2 = 60\\\\`

After 2 minutes, there are 60 gallons left.

`a_n = a_(n-1) - 12\\\\a_3 = a_(3-1) - 12\\\\a_3 = a_(2) - 12\\\\a_3 = 60 - 12\\\\a_3 = 48\\\\`

After 3 minutes, there are 48 gallons left.

`a_n = a_(n-1) - 12\\\\a_4 = a_(4-1) - 12\\\\a_4 = a_(3) - 12\\\\a_4 = 48 - 12\\\\a_4 = 36\\\\`

After 4 minutes, there are 36 gallons left.

`a_n = a_(n-1) - 12\\\\a_5 = a_(5-1) - 12\\\\a_5 = a_(4) - 12\\\\a_5 = 36 - 12\\\\a_5 = \boldsymbol{24}\\\\`

After 5 minutes, there are 24 gallons left.

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Table of values

`\begin{array}c \cline{1-2}n & a_n\\\cline{1-2}0 & 84\\\cline{1-2}1 & 72\\\cline{1-2}2 & 60\\\cline{1-2}3 & 48\\\cline{1-2}4 & 36\\\cline{1-2}5 & \boldsymbol{24}\\\cline{1-2}\end{array}`