a1 = 0.1 a2 = 0.10 a3 = 0.100 a4 = 0.1001 a5 = 0.10011 . . . .Does this sequence converge or diverge? Prove your assertion.
If you said 10*12*5=$600, you're either forgetting that you get
paid interest, or you may want to change your financial institution!
Suppose now that your money is invested in an account with the
Associated Central Credit Union of Mathematics Undergraduate Learners
And Technology Engineers (ACCUMULATE). How much would you have in your
account at age 20 if there is an annual rate of 6% paid monthly?
Scratchwork:
"During the first year, I save up 10*12=$120. After interest,
there is 120*(1.06)=$127.20 at the end of the second year, plus
another $120 I've saved during the second year, so I'll have
$247.20 after two years.
"By the end of the third year, this will grow to $262.032 with the
6% interest, plus the $120 I've saved during the third year, so
I'll have $382.032 at the beginning of the 4th year.
"By the end of the 4th year, this will grow to $404.95 from the
interest, plus $120 saved during that year, giving me $524.95 at
the beginning of my 5th year.
"By the end of the 5th year, this grows to $556.48 from the
interest, and adding the $120 I will have saved during the 5th
year, I'll have $676.48"
What's wrong with this?_______________________________________________.
Why does doing things the right way seem daunting?____________________
___________________________________________________________________.
Now consider the expression
Call this expression E. Multiply E by (1-r) and simplify.
What do you get? ____________________. Hence E = ____________________.
Suppose now that you started depositing $10 each month starting at age 15 and received 0.5% interest per month. When componded monthly, this will add up to more than 6% per year (hence the term "annual percentage yield"). How much more? Make a chart of how much money you have in your account at different times during your first few months.
Here t represents time in months, where t=0 represents the moment you make your first deposit, and r is the monthly interest multiplier of 1.005.
------------------------------- | t (in months) | Savings (in $)| Scratchwork: |---------------|---------------| | 0 | 10 | | | | | 1 | 10*r + 10 | | | | | 2 | _____________ | | | | | 3 | _____________ | | | | -------------------------------Your savings at age 20 will total:____________________ (use your formula!)
When you're 20 years old, you are able to save $25 per
month. Given the same ACCUMULATE interest rates, how much will you
have when you're 25?
When you're 25 you're able to save $250 per month. how much will
you have when you're 40? When you're 50? When you're 60?
Write down a general formula to help your friends and relatives
make similar calculations. Suppose your friend gets a rate of return of
r (where r>1) each month, and puts in $D each month
for Y years. Suppose they also started with $M in their
account at the beginning of the period. How much money will they have
after Y years?
What conditions must be true for r for this fomula to be valid? ____________________.
Otherwise, if _r________ then _____________.
(A) What do you think
equals?______. Confirm this with the above formula:
What modification did you need to make?______________________________.
Can you also verify this formula with a picture?
(B) Use your formula to compute
What equation do you get? ____________________________________________.
Now, multiply both sides of your equation by 9.
What have you proved?_____________________________________________.
What is (1/4) + (1/4)2 + (1/4)3 + ... ?
(This is another special series which (unlike most) can be verified
with a picture as well).