Aron is training for a 10 km marathon swim in 19 weeks. Each week, his training plan includes a long distance swim. In week six, Aron swam a total of 18.75 km. In week ten, Aron swam a total of 36.25 km. Each week, Aron increased the distance he swam following an arithmetic sequence until the final week. In the last week, Aron only swam 5 km in order to save energy for the race.
a. By what distance did Aron increase his total swimming each week?
b. Determine the distance Aron swam in the first week of training.
c. Determine how far Aron swam in the 18-week training program
6 years ago
Answered By Faizan K
a) 4.375 km
b) -3.125
c) 71.25
6 years ago
Answered By Nitin G
Since Aron followed an arithmetic progression, he would have increased the distance he swam by a constant number every week.
Let us assume he swam a kms in first week and increased the distance by d kms every week.
in week 6, he swam 18.75 km.
formula for 6th term is a + 5d
so, a + 5d = 18.75 ................. equation 1
similarly in week 10, he swam 36.25 km
similarly, a + 9d = 36.25 ................. equation 2
Solving equation 1 and equation 2
a = 3.125
d = 4.375
so answer for a point is 4.375
so answer for b point is 3.125
so answer for c point = a + 17 d = 77.5
6 years ago
Answered By Harrison V
An arithmetic progression follows the format: an=a1+(n-1)d, where n is the nth term of the sequence, and d is the step value (that is to say, the common difference between terms). All our terms are in kilometers.
We know: a6=18.75, a10=36.25. We lack the values a1 and d. Nevertheless, we have a solvable system of equations.
a6=a1+5d=18.75
a10=a1+9d=36.25
We take the second equation (or the first, either is acceptable) and set it equal to a1, like so,
a1=18.75 -5d
and we plug a1 into our second equation,
18.75 -5d+9d=36.25
4d=36.25-18.75=17.5
d=17.5/4=4.375
Question a). 4.375
Now we determine the value of a1 by plugging in d to one of our equations.
a6=a1+5(4.375)=18.75
a1=18.75-5(4.375)=-3.125
Which is strange because I don't know how he could swim negative distance, but I am entirely confident in the answer so we'll accept the minus sign when giving the answer to Question B. However, I believe it is best to keep the minus sign in the mathematics, since that's what previous values require (even if it's not really a physically possible answer).
Question b). a1=-3.125
It sounds like question c is asking for the total distance traveled. With some googling we can easily find the formula that determines the sum of an arithmetic sequence.
$sum=n\left(\frac{a1+an}{2}\right)$sum=n(a1+an2 )
a18=-3.125+17(4.375)=71.25
We have n=18, a18=71.25
$sum=18\left(\frac{-3.125+71.25}{2}\right)=613.125km$sum=18(−3.125+71.252 )=613.125km
Question c). He swam 613.125km in total.