Math Problem Solver

. Start with a positive integer, then choose a negative integer. We’ll use these two numbers to

generate a sequence using the following rule: create the next term in the sequence by adding

the previous two. For example, if we started with 6 and −5, we would get the sequence

6, −5, 1, −4

| {z }

alternating part

, −3, −7, −10, −17, −27, . . .

which starts with 4 elements that alternate sign before the terms are all negative. If we started

with 3 and −2, we would get the sequence

3, −2, 1, −1

| {z }

alternating part

, 0, −1, −1, −2, −3, . . .

which also starts with 4 elements that alternate sign before the terms are all non-positive (we

don’t count 0 in the alternating part).

(a) Can you find a sequence of this type that starts with 5 elements that alternate sign?

With 10 elements that alternate sign? Can you find a sequence with any number of

elements that alternate sign?

(b) Given a particular starting integer, what negative number should you choose to make

the alternating part of the sequence as long as possible? For example, if your sequence

started with 8, what negative number would give the longest alternating part? What if

you started with 10? With n?