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• Run Multiple Choice Generator at h:\shares\physics
• Run Core Problems by giving the program 163 as your 'course'--enter this when asked for your 3-digit course number.  
• Do choice #3, then do choice #9.
• If you miss a problem be sure to click on Comment and explain what you do and do not understand about the question and the given solution

Problem Number 7

If a(n) = a(n-1) + 9, with a(0) = -9, then what is the value of a( 310)?

Compounding $1000 initial principle continuously, for 1 year, at 100% annual interest:

• If we compound just once we end up with $2000, which is obvious.
• If we compound twice then we get to add 50%, then add another 50% to the resulting amount.  Adding 50% is the same as multiply by 1.5  So we multiply the initial principle by 1.5, then multiply the result by 1.5.  We end up with $1000 * 1.5 * 1.5 = $1000 * 1.5^2 = $2250.
• If we compound, say, 100 times we get to add 1%, then add another 1% to the resulting amount, etc., 100 times.  Adding 1% is the same as multiply by 1.01  So we multiply the initial principle by 1.01, then multiply the result by 1.01, etc., 100 times.  We end up with $1000 * 1.01^100 = $2704, a factor of 2.704.
• If we compound n times then we get to add 100% / n, a total of n times. 
• Adding 100% / n = 1.00 / n or just 1/n is the same as multiplying by 1 + 1 / n.  So we multiply the initial principle by 1+1/n, then multiply the result by 1+1/n, etc., n times. 

• We end up with $1000 * (1 + 1/n) ^ n.
• As n -> infinity compounding approaches continuous and (1 + 1/n) ^ n approaches e.  e is approximately 2.71828.

If we compound interest annually at annual rate 8% then if we start with $1000, how much do we have after 5 years?

Compounding annually we multiply our principle by 1.08 every year, so we end up with $1000 * 1.08^5 = $1469.32.

If we compound interest monthly at annual rate 8% then if we start with $1000, how much do we have after 5 years?

Compounding monthly we multiply our principle by 1 + .08 / 12 = 1.006667 every month.  In 5 years we do this 5 * 12 times, so we end up with $1000 * 1.006667^(5 * 12) = $1487.90.

If we compound interest continuously at annual rate 8% then if we start with $1000, how much do we have after 5 years?

Continuous compounding gives us principle function P(t) = $1000 * e^(.08 t).

For t = 5 we have principle

• P(5) = $1000 * e^(.08 * 5) = $1489.82

The most basic exponential function forms are

• y = 2^x and
• y = e^x

The generalized forms for horizontal asymptote zero are

• y = A * 2^(kx)
• y = A * e^(kx)
• y = A * b^x.

Using the form y = A * b^x, assume that y = 20 when x = 5 and y = 30 when x = 12.  Find the values of the parameters A and b.