Question

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 1

Let
`a_n` be the n'th term of a sequence,

for example
`a_1` is the first term,
`a_2` is the second term and so on.
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a sequence is arithmetic if the difference between any 2 terms is equal:

so an arithmetic sequence, for the first term = t, and the common difference = d, has the following form:


`a_1=t`


`a_2=t+d`


`a_3=t+d+d`


`a_4=t+d+d+d`

so clearly
`a_n=t+d*(n-1)`
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A sequence is geometric, if the ratio between any 2 consecutive terms is the same, and it is called the common ratio.

a geometric ratio with first term = t, and common ratio = r is:


`a_1=t`


`a_2=t*r`


`a_3=t*r*r`


`a_4=t*r*r*r`

thus clearly
`a_n=t* r^(n-1)`
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Part 1:

"Jasmine practices the piano for __30___ minutes on Monday. Every day she ____increases_______ her practice time by ____5 minutes_____. "


let
`a_n` be the number of hours Jasmine practices the n'th day,

for n=7, t=30, d=5 we have


`a_7=30+5*(7-1)=30+5*6=30+30=60`,

minutes is the time Jasmine practices on the 7the day.


Part 2

"Anthony goes to the gym for ___60___ minutes on Monday. Every day he ___increases______his gym time by ____1/10_of the previous day_____. "

if
`S_n` represents the minutes that Anthony goes to gym on the n'th day,

then


`S_1=60`


`S_2=60+1/10*60=10/10*60+1/10*60=11/10*60=1.1*60`


`S_3=1.1*1.1*60`


`S_4=1.1*1.1*1.1*60`


`S_5=1.1*1.1*1.1*1.1*60= (1.1)^(4)*60= 1.4641*60=87.85`
(minutes)

Part 3: it is clear that the formula that can be derived in Part 2 is:


`S_n=60*(1.1)^(n-1)`

so assume we want to find how many hours does Anthony spend in the gym on the 10th day,

then we calculate
`S_10=60*(1.1)^(9)=60*2.358=141.5` (minutes)