Question

Alan and Greta will be competing in a marathon that will be held in 15 weeks, Alan and Greta decide to start training for a marathon. Alan decides to start preparing for the marathon by running 12 hours during the first week and will increase by 2 hours each week. Greta has decided to start training by running 12 hours in the first week and will increase her time by 10% each week.
How many hours per week will each person run in the 5th week?

Answer 1

i know you have to use the arithmetic sequence for alan n the geometric sequence for greta.

Answer 2

@SolomonZelman @Callisto @Compassionate @mathslover

Answer 3

Hmm ok this is kind of a fun one :)

Answer 4

I think it will help us to keep track of the terms if we label them well.
So let's call Alan's terms a bunch of A's.

Answer 5

The first week, Alan ran for 12 hours:\[\Large\rm A_1=12\]The next week he ran for 2 extra hours:\[\Large\rm A_2=12+2\]Mmm ok good you've got the right idea.
Arithmetic for Alan, yah?

Answer 6

Notice that we didn't add a 2 the first time.
So we're always adding one less 2 than the term number that we're on.
So for the 5th term, we will have added four 2's.

Answer 7

\[\Large\rm A_5=12+4(2)\]yah?

Answer 8

How bout Greta, any ideas?

Answer 9

hold on im still trying to figure this out

Answer 10

ok i got n for greta i know you use the geometric sequence, but idk how i do it

Answer 11

So like.. she starts at 12.\[\Large\rm G_1=12\]

Answer 12

And increases 10% each term.

Answer 13

So think of it this way, the next term will be 100% and 10% more in comparison to the previous term, yes?
That's what it means to increase by 10%, we'll be at 110% for the next term.

Answer 14

But we would like the percent to be a decimal value so we can use it in our multiplication.\[\Large\rm 110\text{%} \quad\to\quad 1.1\]

Answer 15

\[\Large\rm G_1=12\]\[\Large\rm G_2=12(1.1)\]

Answer 16

That's 100 and 10 percent of the 12. ^
Which signifies an increase of 10%.

Answer 17

The third week would be another increase of 10%,\[\Large\rm G_3=12(1.1)(1.1)\]\[\Large\rm G_3=12(1.1)^2\]

Answer 18

ok so for week five it would be, G= 12(1.1)^4

Answer 19

Right? @zepdrix

Answer 20

Yes very good! :)

Answer 21

thnx cx