Please help with arithmetic and geometric sequences!!

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Please help with arithmetic and geometric sequences!!

Mathematics
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Part 1: Create a scenario for an arithmetic sequence. For example, Jasmine practices the piano for ______ minutes on Monday. Every day she ___________ her practice time by _________. If she continues this pattern, how many minutes will she practice on the 7th day? Be sure to fill in the blanks with the words that will create an arithmetic sequence. Use your scenario to write the function for the 7th term in your sequence using sequence notation. Part 2: Create a scenario for a geometric sequence. For example, Anthony goes to the gym for ______ minutes on Monday. Every day he _________his gym time by ____________. If he continues this pattern, how many minutes will he spend at the gym on the 5th day? Be sure to fill in the blanks with the words that will create a geometric sequence. Use your scenario to write the formula for the 5th term in your sequence using sequence notation. Part 3: Use your scenario from part 2 to write a question that will lead to using the geometric series formula. Use the formula to solve for Sn in your scenario.
I do not know the difference between a geometric or arithmetic sequence that well, so I have no work for this question...

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Other answers:

  • phi
arithmetic means you go up by a fixed amount. the simplest "arithmetic sequence" is 1,2,3,... but you could go up by any fixed amount. for example: 0.4 0, 0.4, 0.8, 1.2, ...
  • phi
geometric means you *multiply* by a fixed value for example, if we start at 1, and multiply each term by 2 to get the next term, the sequence would be 1,2,4,8,16... if you multiply by a number less than 1, the sequence gets smaller for example, if we start at 8 , and multiply by 1/2, we get 8,4,2,1, 1/2 , 1/4 ...
  • phi
If you want more background, see https://www.khanacademy.org/math/algebra/sequences/arithmetic_sequences/v/arithmetic-sequences
Ok, thanks, let me watch that and start working on them. Be right back. Then I'll show you my work.
So, for the arithmetic sequence, I can start with any number, and then I have to have it go up by the same amount every time?
  • phi
yes. For practice time, I would pick a positive number of minutes for the start and a small increase (positive) number to add each day
ok, so I think I'll start with 15 minutes, and add 5.
  • phi
ok
  • phi
then write the formula \[ a_n = a_1 + (n-1) d \] where \( a_1 =15 \) is the starting value and d is the amount you increase by = 5
ok
For part 2, how should I choose numbers to make it a geometric sequence?
  • phi
did you finish part 1. what is \(a_7\) ?
it's 45, right?
  • phi
can you show your work?
  • phi
\[ a_7= 15 + (7-1) \cdot 5 \\ = 15+6\cdot 5 \\=15+30 \\=45\]
ok, that's what I have
  • phi
For part 2, I would increase his time by (for example) 10% which means multiply by 1.1 or (a bit unrealistic) you can say increase his time ... by a factor of 2 which means multiply by 2
ok
I think I will do a factor of 2, as that will be an easier number to work with
  • phi
Use this formula to find the "nth term" (n=5 for your problem) \[ a_n= a_1 \cdot r^{(n-1)} \]
  • phi
people usually use r as the factor we multiply by. in your problem you are using r=2
ok, let me work on that
  • phi
For part 3 here is the a video on the formula to add up the terms in a series https://www.khanacademy.org/math/precalculus/seq_induction/geometric-sequence-series/v/geometric-series (it may be too advanced?) a reasonable question would be how many minutes did Anthony work out at the gym the first week, (assuming he followed the geometric sequence from part 2) ?
ok, thank you!

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