Question

Eric jogged 3 1/4 on Mon, 5 5/8 >; Tues and 8 miles on Wed. Suppose he continues this pattern for the remainder of the week. how far will Eric jog on Fri?

Answer 1

Sequences

The distance in miles jogged by Eric in the week are shown below:

Mon: 3 1/4

Tue: 5 5/8

Wed: 8

The distances form a pattern which we will recognize below.

Let's compute the differences in the distance of each day with respect to the previous day:

$r=5\frac{5}{8}-3\frac{1}{4}$

Converting each mixed fraction into improper fractions:

$r=(5+\frac{5}{8})-(3+\frac{1}{4})=\frac{45}{8}-\frac{13}{4}$

The LCM of the denominators is 8, thus:

$r=\frac{45}{8}-\frac{13}{4}=\frac{45}{8}-\frac{26}{8}=\frac{19}{8}$

Now we find the same difference between the third and the fourth terms of the sequence:

$r=8-\frac{45}{8}=\frac{64-45}{8}=\frac{19}{8}$

Since both differences have the same value, the terms of the distances jogged by Eric form an arithmetic sequence. The general term of an arithmetic sequence is:

$a_n=a_1+(n-1)\cdot r$

Where a1 is the first term, n is the number of the term, and r is the common difference.

We have the values a1=3 1/4= 13/4. r=19/8, thus the distance jogged by Eric on Friday (n=5) is:

$a_5=\frac{13}{4}+(5-1)\cdot\frac{19}{8}$

Calculating:

$a_5=\frac{13}{4}+4\cdot\frac{19}{8}=\frac{13}{4}+\frac{19}{2}$

The LCM of 2 and 4 is 4, thus:

$a_5=\frac{13}{4}+\frac{38}{4}=\frac{51}{4}$

Converting to mixed fraction:

$a_5=\frac{51}{4}=\frac{48+3}{4}=12+\frac{3}{4}=12\frac{3}{4}$

Eric jogged 12 3/4 miles on Friday