Question

Please help with arithmetic and geometric sequences!!

Answer 1

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.

Answer 2

I do not know the difference between a geometric or arithmetic sequence that well, so I have no work for this question...

Answer 3

@phi

Answer 4

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, ...

Answer 5

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 ...

Answer 6

If you want more background, see
https://www.khanacademy.org/math/algebra/sequences/arithmetic_sequences/v/arithmetic-sequences

Answer 7

Ok, thanks, let me watch that and start working on them. Be right back. Then I'll show you my work.

Answer 8

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?

Answer 9

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

Answer 10

ok, so I think I'll start with 15 minutes, and add 5.

Answer 11

ok

Answer 12

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

Answer 13

ok

Answer 14

For part 2, how should I choose numbers to make it a geometric sequence?

Answer 15

did you finish part 1. what is \(a_7\) ?

Answer 16

it's 45, right?

Answer 17

can you show your work?

Answer 18

\[ a_7= 15 + (7-1) \cdot 5 \\ = 15+6\cdot 5 \\=15+30 \\=45\]

Answer 19

ok, that's what I have

Answer 20

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

Answer 21

ok

Answer 22

I think I will do a factor of 2, as that will be an easier number to work with

Answer 23

Use this formula to find the "nth term" (n=5 for your problem)
\[ a_n= a_1 \cdot r^{(n-1)} \]

Answer 24

people usually use r as the factor we multiply by. in your problem you are using r=2

Answer 25

ok, let me work on that

Answer 26

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) ?

Answer 27

ok, thank you!