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(5)/(8)-3(1)/(4)`

Converting each mixed fraction into improper fractions:

`r=(5+(5)/(8))-(3+(1)/(4))=(45)/(8)-(13)/(4)`

The LCM of the denominators is 8, thus:

`r=(45)/(8)-(13)/(4)=(45)/(8)-(26)/(8)=(19)/(8)`

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

`r=8-(45)/(8)=(64-45)/(8)=(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=(13)/(4)+(5-1)\cdot(19)/(8)`

Calculating:

`a_5=(13)/(4)+4\cdot(19)/(8)=(13)/(4)+(19)/(2)`

The LCM of 2 and 4 is 4, thus:

`a_5=(13)/(4)+(38)/(4)=(51)/(4)`

Converting to mixed fraction:

`a_5=(51)/(4)=(48+3)/(4)=12+(3)/(4)=12(3)/(4)`

Eric jogged 12 3/4 miles on Friday