# A runner targets herself to improve her time on a certain course by 3 seconds a day.

• A runner targets herself to improve her time on a certain course by 3 seconds a day. If on day 0 she runs the course in 3 minutes, how fast must she run it on the 14th day to stay on target?

Q.4b

Let be the sequence defined recursively as follows:

Use iteration to guess an explicit formula for the sequence?
Solution

ak = ak−1 + 2 for all integers k ≥ 1

a = a−1 + 2

a1 = a0 + 2,

a2 = a1 + 2,

a3 = a2 + 2,
and so forth. Now use the initial condition to begin a process of successive substitutions into these equations, not just of numbers (as was done in Section 5.6) but of numerical expressions. The reason for using numerical expressions rather than numbers is that in these problems you are seeking a numerical pattern that underlies a general formula. The secret of success is to leave most of the arithmetic undone. However, you do need to eliminate parentheses as you go from one step to the next. Otherwise, you will soon end up with a bewilderingly large nest of parentheses. Also, it is nearly always helpful to use shorthand notations for regrouping additions, subtractions, and multiplications of numbers that repeat. Thus, for instance, you would write

5·2 instead of 2 + 2 + 2 + 2 + 2

Notice that you don’t lose any information about the number patterns when you use these shorthand notations. Here’s how the process works for the given sequence:

a0 = 1
a0=1
a1=a0+2=1+2
a2=a1+2=(1+2)+2 =1+2+2
a3=a2+2=(1+2)+2 =1+2+2+2
a4=a3+2=(1+2+2+2)+2=1+2+2+2+2
Since it appears helpful to use the shorthand k ·2 in place of 2 + 2 +···+ 2 (k times), we do so, starting again from a0.

a0 = 1 = 1 + 0·2
a1 = a0 + 2 = 1 + 2 = 1 + 1·2
a2 = a1 + 2 = (1 + 2) + 2 = 1 + 2·2
a3 = a2 + 2= (1 + 2·2) + 2 = 1 + 3·2
a4 = a3 + 2 = (1 + 3·2) + 2 = 1 + 4·2
a5 = a4 + 2 = (1 + 4·2) + 2 = 1 + 5·2

guess, an = 1 + n ·2 = 1 + 2n

The answer obtained for this problem is just a guess. To be sure of the correctness of this guess, you will need to check it by mathematical induction. Later in this section, we will show how to do this.

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