On the first day, 42 liters of water are poured into a tank. Every day thereafter, three times as much water is poured into the tank as was poured on the previous day. On which day are 1134 liters poured into the tank?
Let’s think about what’s really happening here. On the first day, we have 42 liters in the tank. On the next day, three times as much water is poured into the tank as was poured in the previous day. That’s 42 times three, which is 126 liters. Then, on the third day, three times the amount of water are poured into the tank again. That’s 126 times three, which is 378 liters. So, we’ve generated a sequence. In fact, this sequence has a special name. It’s a geometric sequence. This is one where each term is found by multiplying the previous term by a fixed nonzero number that we call the common ratio.
Now, in this case, our common ratio, which we’ll define to be 𝑟, is three. We’re multiplying the amount of water poured into the tank each day by three. We can also define the first term in our sequence to be 𝑎, and that’s equal to 42 or 42 liters. And so, we’re then able to recall the 𝑛th term for a geometric sequence. It’s 𝑎 sub 𝑛 equals 𝑎 times 𝑟 raised to the power of 𝑛 minus one.
Now, the question wants us to find out which day 1134 liters are poured into the tank. In other words, which value of 𝑛 yields 𝑎 sub 𝑛 is equal to 1134? Let’s substitute 𝑎 sub 𝑛 as 1134, 𝑎 as 42, and 𝑟 as three into our formula for the 𝑛th term. Remember, we don’t know which value of 𝑛 this is true for. So, we write 1134 equals 42 times three to the power of 𝑛 minus one. We’re looking to solve for 𝑛. So, let’s divide through by 42. 1134 divided by 42 is 27.
So, we see that three to the power of 𝑛 minus one is equal to 27. But of course, we know that three cubed is 27. And so, for this statement to be true, for 27 to be equal to three to the power of 𝑛 minus one, 𝑛 minus one must actually be equal to three. And that’s an equation we can solve for 𝑛 by adding one to both sides. So, we see 𝑛 is equal to four.
And so, we can say that it’s day four on which 1134 liters is poured into the tank. And, of course, for such a small value of 𝑛, we could have simply continued our sequence by multiplying each term by three. Had we, in fact, done so, we would have found that the term after 378 is 1134 as required.