Arithmetic & geometric progressions

Nov 2016
5
0
England
Can anyone help with these?
8. A small company producing children’s toys plans an increase in output. The number of toys produced is to be increased by 25 each week until the weekly number produced reaches 5000. In week 1 the number to be produced is 380.

a. How many weeks will it take to reach maximum output?

b. What is the total number of toys produced over the first 52 weeks?


9. A post is being driven into the ground by a mechanical hammer. The distance it is driven by the first blow is 11cm. Each subsequent blow drives the post 90% of the distance of the proceeding blow.

a. If each blow takes 10 seconds how long will it take to drive the post at least 75cm into the ground

b. What is the maximum distance the post can be driven into the ground?
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
Post moved to new thread ... for the OP, do not piggy-back a new problem onto an older thread. Thank you for your cooperation.
 
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Nov 2016
5
0
England
Will bear in mind, apologese!
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Also it is hard to believe that you know absolutely nothing about how to do these problems!

"In week 1 the number to be produced is 380. The number of toys produced is to be increased by 25 each week until the weekly number produced reaches 5000."
So the number produced has to go from 380 to 5000. How large an increase is that? If there is an increase of 25 each week, how many weeks will that require?

"What is the total number of toys produced over the first 52 weeks?"
This a little bit harder. You titled this "Arithmetic and geometric progressions" so I assume you recognize this as an "arithmetic progression" with starting value 380 and "common difference" 25. Do you know a formula for the sum of an arithmetic progression?

"9. A post is being driven into the ground by a mechanical hammer. The distance it is driven by the first blow is 11cm. Each subsequent blow drives the post 90% of the distance of the proceeding blow."
Do you recognize this a geometric progression with starting value 11 and "common ratio" 0.9?

"a. If each blow takes 10 seconds how long will it take to drive the post at least 75cm into the ground"
Again, there is a formula for the sum of a finite geometric progression. Do you know it?

"b. What is the maximum distance the post can be driven into the ground?"
This can be modeled as an infinite geometric progression. Do you know a formula for the sum of an infinite geometric progression?
 
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Nov 2016
5
0
England
Firstly, I acquired a C in maths GCSE over 10 years ago and have never needed to look at problems like I have posted since. Secondly I have been shoved onto a HNC Mechanical Engineering course through my employers, which I am not arguing about I mean its a good qual but I was always going to struggle with the maths, hence using this forum! However I am determined to pass and have fully applied myself!
Since I first posted I have had 2 lessons and Ive a much better idea of how to solve these problems:

8 a. How many weeks will it take to reach maximum output?

I found variables; a = 380, d = 25, n = ?, Sn = 5000
Therefore I used the formula: n = a + (n-1) x d. But essentially in reverse, I think!
5000 - 380 = 4620 / 25 = 184.5
So if n - 1 = 184.5 then n must equal 185.8 right?

b. What is the total number of toys produced over the first 52 weeks?
Your right this one was more tricky, but i went with this:

Again, variables: a = 380, d = 25, n = 52, Sn = ?

Then the formula; Sn = n/2 (2 x a + (n - 1) x d)
So, 52/2 (2 x 380 + (52 - 1) x 25 = 53,560 = Sn

Some of my peers have made this answer different slightly, but I don't see how?

9 a. If each blow takes 10 seconds how long will it take to drive the post at least 75cm into the ground?
a = 11, r = 0.9, Sn = 75
Sn = a (r^n - 1) / r - 1 (shown as fraction)
75 = 11 (0.9^n - 1) / 0.9 - 1
0.9 - 1 x 75 = -7.5
7.5 = 11 (0.9^n - 1)
Then divide either side by 11
-7.5 / 11 = -0.681 then +1 as both sides have been treated the same = 0.318
0.318 = 0.9^n
Using Logarithm functions:
log 0.9 (0.318) = 10.874 x 10 (seconds) = 108.75 (109)seconds

b. What is the maximum distance the post can be driven into the ground?

Correct me if Im wrong but i believe this answer to technically be infinite, but I still get a figure I can use.
a = 11, r = 0.9
S (infinite symbol) = a / 1 - r
11 / 0.1 = 110cm.

If your not fed up of me by now, I have a few more questions that I need help with!
 
Nov 2016
5
0
England
To start, Mr. Google can:
search "arithmetic and geometric series";
lots of teachers will pop up...
Yes but to someone who has never seen or used these series, google further confused me. I felt a human would be better help, but THANKS
 
Feb 2015
2,255
510
Ottawa Ontario
Yes but to someone who has never seen or used these series, google further confused me. I felt a human would be better help, but THANKS
Look at your initial post: you gave NO indication of where you're at.
Even looks like you were trying to get your homework done; agree?
 
Nov 2016
5
0
England
Not homework, assignment questions. If I just wanted answers then yes Google is there, but I need methods which is why I'm here. Thanks for all your help Dennis, your a real swell guy