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John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. . Explain why this series is convergent or divergent

- anonymous

- katieb

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- anonymous

just tell me is this convergent or divergent

- SolomonZelman

well, just paying 1.1 times the previous payment means that sequence diverges (nothing to even talk about a series).
BUT, a loan (I would think so) is some finite amount though, so this situtation with a loan is a bad example.....

- SolomonZelman

he won't go into paying more than the loan...... (and this all I am assuming there is no interest, since it is not mentioned here)

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## More answers

- SolomonZelman

So, lets write this series, and lets diregard the fact that he is to pay some finite loan amount. OK?

- anonymous

is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

- anonymous

is divergent

- SolomonZelman

\(\large\color{black}{ \displaystyle a_1=100 }\)
\(\large\color{black}{ \displaystyle r=1.1 }\)
and this is a perfect geometric series.
For geometric series to converge, normally, the folloing mst be true: `|r|<1`
yes correct (disregarding the fact that it's a loan).

- SolomonZelman

The fact that it is a loan is forcing a FINITE series.

- SolomonZelman

If you were to propose a different scenario, such as:

- SolomonZelman

John is giving a charity to poor children who stand by the Church. THe first month he gave 100$ and then, every month, he wants to give 1.1 times as much as he gave the previous month. is this series convergent or divergent?

- SolomonZelman

this would be more like an INFINITE series scenraio....

- anonymous

ok thanks can u help with another question

- SolomonZelman

ok, but what course is this?

- SolomonZelman

is this geometry or calc?

- anonymous

algebra 2

- SolomonZelman

oh, i see.... go ahead...

- SolomonZelman

((in calc there are more advanced "tools" to use))

- anonymous

The graph of f(x) = 2^x + 1 is shown below. Explain how to find the average rate of change between x = 0 and x = 3

- SolomonZelman

find f(0).
find f(3).
find the slope between (0,f(0)) and (3,f(3)).

- SolomonZelman

making sense?

- anonymous

##### 1 Attachment

- anonymous

thats the graph

- SolomonZelman

no need for a graph to perform this task, but, it is nice to have one.

- SolomonZelman

do my steps to the solution make sense?

- anonymous

no

- SolomonZelman

Ok, lets do it togther one by one.

- anonymous

so basically your finding slope

- SolomonZelman

yes

- SolomonZelman

did you find f(0) and f(3) ?

- SolomonZelman

f(x) = 2\(^x\)+1

- SolomonZelman

f(0)=2\(^0\)+1=?
f(3)=2\(^3\)+1=?

- anonymous

f(3) =7

- SolomonZelman

close

- SolomonZelman

2Â³ = ?

- anonymous

9

- anonymous

oops

- SolomonZelman

9 is f(3) or just the 2Â³ part?

- anonymous

2^3

- SolomonZelman

\(2^3=2\times 2 \times 2 \)

- SolomonZelman

=8

- anonymous

f(0) = 1

- anonymous

f(3) = 9

- anonymous

i meant that

- SolomonZelman

yes,

- SolomonZelman

f(3)=9

- SolomonZelman

and f(0)=?

- anonymous

1

- SolomonZelman

no

- SolomonZelman

your function is
\(f(x)=2^x+1\)

- SolomonZelman

\(f(0)=2^0+1=?\)

- anonymous

f(0) = 0 + 1

- SolomonZelman

\({\rm C}^0=1\) (for any real number C)

- anonymous

f(0) = 1

- SolomonZelman

\( f(0)=2^0+1=1+1=?\)

- anonymous

2

- SolomonZelman

yes, f(0)=2

- anonymous

Explain how to find the average rate of change between x = 0 and x = 3.

- SolomonZelman

now, you need to find the slope between (0,f(0)) and (3,f(3)).
we know f(3)=9, f(0)=2. SO lets plug this in.
~ you have to find the slope between (0,2) and (3,9)

- SolomonZelman

\(\LARGE \color{black}{ \displaystyle {\rm m}=\frac{\color{blue}{{\rm y}_1}-\color{red}{{\rm y}_2}}{\color{green}{{\rm x}_1}-\color{darkgoldenrod}{{\rm x}_2}} }\)
where
\(\LARGE \color{black}{ \displaystyle {\rm m} }\) is the slope
\(\Large\color{black}{ \displaystyle (\color{green}{{\rm x}_1}~,~~\color{blue}{{\rm y}_1}) }\) and \(\Large\color{black}{ \displaystyle (\color{darkgoldenrod}{{\rm x}_2}~,~~\color{red}{{\rm y}_2}) }\) are your two points.

- anonymous

9-2 =7
3-0 =3

- SolomonZelman

yes, so your slope is?

- anonymous

7/3

- SolomonZelman

yup

- anonymous

is that the final answer

- SolomonZelman

yes, that is the final answer for your second question.

- anonymous

so what is the average rate of change again

- SolomonZelman

7/3 :)

- SolomonZelman

slope and average rate of change (in this context) is the same thing.

- anonymous

thanks so much

- SolomonZelman

you are welcome !

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