We have:

a1 = 200
a2 = 240

and the sequence is constantly increasing (we are adding 200 to larger and larger values the further we get into the sequence), so a50 must be greater than 240. In addition, we find the next term (a_N) in the sequence by adding 20% of the previous term (a_(N-1)) to 200. Since we do not yet have a term in our sequence which is 250 or greater, we will never add a number as large as 0.2*250 = 50 to 200 to make a term - there will therefore never be a term as large as 250 in our sequence. The values will, however, get very close to 250. So after a2, every term is somewhere between 240 and 250.

Uri wrote:

Ian Stewart wrote:Since we do not yet have a term in our sequence which is 250 or greater, we will never add a number as large as 0.2*250 = 50 to 200 to make a term - there will therefore never be a term as large as 250 in our sequence. The values will, however, get very close to 250. So after a2, every term is somewhere between 240 and 250.

Stewart, although I understand that the terms are gradually getting closer to 250, could you please explain the fundamental on which you told that the term will never cross 250?

I can try:

Say our sequence is a, b, c, d, e, f, g, ...

We know that we find the next term of our sequence by adding 20% of the previous term to 200. So if we know a, we'd find b by adding 20% of a and 200. Now, if a is less than 250, 20% of a must be less than 50, so b will be less than 250. If b is less than 250, then 20% of b will be less than 50, so c will be less than 250. And so on...

Hope that makes sense - it's probably easier to see if you start with 200 and find the next two or three terms; working out the first few terms is almost always a good idea in sequence questions.