SOLUTION: can you help. I am trying to help my son with his algebra homework. His school does not send books home and I haven't used some of this in 20 years. The problem states. Write

I think you need a more detailed explanation than the other tutor gave,
although his is correct.

First note that those n's are SUBscripts, not exponents (which are SUPERscripts).

The numbers -23,38,99,160 means:

FIRST TERM = 
SECOND TERM = 
THIRD TERM = 
FOURTH TERM = 

In math there are two ways to handle a multiple choice question.

Method 1.  Do the problem as you were taught and mark the choice
           that corresponds to your answer. 

Method 2.  Go through the choices and find the one that satisfies
           all the conditions given in the problem.

The instructor, of course, hopes that the student will use Method 1. 
if the test is objective, then the student must do the problem as
taught.  However in multiple choice exams, when the student is not 
sure how to do the problem, Method 2 will often, but not always, 
lead to the correct answer. This is a case where Method 2 will work!
So I will begin with Method 2, then I will use Method 1.

First try choice a.  to see if  by 
substituting :

         




That works, but now we must make sure it works when n=2, 3, and 4.
So we see if  by substituting 

         




So we see that gives 15, not 38, so we can rule out choice a.

Next try choice b.  to see if  by 
substituting  to see if :

         




So we see that gives 15, not 38, so we can rule out choice b
just by checking only 

Next try choice c.  to see if  by 
substituting :

         




That works, but now we must make sure it works when n=2, 3, and 4.
So we see if  by substituting 

         




That works, too, but now we must make sure it works when n=3 and 4.
So we see if  by substituting 

         




That works, too, but now we only need to make sure it works when n=4.
So we see if  by substituting 

         




So now we know for sure that c is the correct choice.  We don't need to
try d, but if we do, we'll find that it gives 
and so is quickly eliminated. 

Next we'll do it by Method 1, the way your son was taught, although 
you should tell him to use Method 2 on multiple choice exams when he 
is not sure how to start.

Method 1.

First we determine whether this is a geometric or an arithmetic sequence.
[20 years ago these were called "progressions", but in modern times, they
are called "sequences".]

If it is a geometric sequence, then 

, the common ratio.

If it is arithmetic, then 

, the common difference.

For it to be geometric,





This is certainly FALSE, so we know that it is not geometric.

For it to be arithmetic,






This is certainly TRUE, so we know that it is an arithmetic sequence.

Your son was taught formulas for the nth terms of both types of 
sequences.  He was taught that the formula for the nth term of
an arithmetic sequence is



So we substitute  and 






 
And using Method 1 we see that the correct choice is c.

[In case he has other sequence problems that are geometric,
the formula for the nth term is ].

Edwin