Word Problems

Sequences, especially arithmetic and geometric ones, are good for word problems.

Sequence story problems come in two main flavors. If these flavors were ice cream, they'd be vanilla and rocky road. We may have to find

• the value of a particular term a_n. This is the standard vanilla problem.
    
• the value of n at which the terms do something in particular. This is the more complicated, rocky road problem.

In general, it's good strategy to write out the first few terms of the sequence in question so we can see the pattern of the terms. Maybe we can do it with ice cream cone in hand.

Sample Problem

An old story goes that a peasant won a reward from the king, and asked for rice: one grain to be placed on the first square of a chessboard, two grains on the second square, four on the third square, and so on. Each square was to contain double the number of grains on the previous square.

1. How many grains of rice would be on the 32nd square?
     
2. Which square would contain exactly 512 grains of rice?

Answer.

If we look at the first few terms, we can see the pattern:

The n^th square contains 2^{n – 1} grains of rice. We have

a_n = 2^{n – 1}

where n is the square and a_n is how many grains of rice are on that square. Now we're prepared to answer the questions.

1. The question "How many grains of rice would be on the 32nd square?" is asking us to find the value of the 32nd term. No problem:
   
   a₃₂ = 2³¹ = 2,147,483,648.
    
2. The question "Which square would contain exactly 512 grains of rice?" is asking us what value of n makes a_n = 512. We use the formula we have for a_n and solve for n:
   
   512 = a_n
   = 2^{n – 1}
   log ₂ 512 = n – 1
   9 = n – 1
   n = 10
   
   This means the 10th square would contain exactly 512 grains of rice.

Be Careful: One type of sequence problem asks us to find a value of a_n. Another asks us to find a value of n. We should be sure to provide the correct information in our answer.

Sometimes when we're asked to find a value of n, we might solve an equation or inequality and get a value of n that isn't a whole number. This is where the road can get bumpy, so we could take a lick of our rocky road cone, and then we'd try out the whole numbers to either side and see which gives a better answer.