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Confused on compound interest? This should do the trick!

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Compound interest introduction | Interest and debt | Finance & Capital Markets | Khan Academy

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John is a jigglypuff breeder. On the first day of his job, he received 2 jigglypuffs. On the second day, he had 4 jigglypuff, on the third day, he had 8 jigglypuff, on the fourth day, he had 16 jigglypuff, and on the fifth day, he had 32 jigglypuff. Is the average rate of change constant? What type of function is this?

1) Find the average rates of change between each day.

2, 4, 8, 16

2) Since the average rate of change is NOT constant, this is an exponential function.

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y = y(r)^x - x1

Pros: Easily find common ratio; starting value can be any point

Cons: More complicated; might have mistakes with +/-; harder to find y-int

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f(1) = b, f(n) = f(n-1) x r

Pros: Easily find common ratio; identify y-int; helps find next point

Cons: Complicated equation; hard to find next number if you don't know previous one

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y = a(r)^x

Pros: Easy to plug in values; a = y -int; easily find common ratio

Cons: Hard to write with if NOT given starting value

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f(1) = b, f(n) = f(n-1) + d

Pros: Easy to find next value if you know previous one; gives y-int; gives slope

Cons: Lots of numbers and letters; deals with function notation; can't find ANY point if previous is unknown

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ax + by = c

Pros: Find x and y ints easily; adding two values of two variables makes it easy to graph

Cons: Harder to find slope; have to do work to find intercepts

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y = m (x - x1) + y1

Pros: Shows slope; gives point on graph

Cons: Doesn't show starting point; easy to make mistakes graphing with +/-; lots of numbers

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y = mx + b

Pros: Shows y -int and slope; easy to graph

Cons: If the y-int is not given it's hard to graph; if there are two variables it's confusing to graph

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A = Prt

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A = Amount of money in account after t years

P = Principle (initial amount deposited)

r = interest rate

t = time

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A = P(1 -r) ^t

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A = P (1 + r) ^t

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To find the average rate of change between two points:

f(a) - f(b)
---------------
a -b

Note: If the function is linear, the average rate of change is the slope and remains constant. If it's exponential, the average rate of change only describes two points, and is NOT constant.

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Increasing at a constant rate: Average rates are the same (positive slope)

Increasing at an increasing rate: Average rates are an increasing linear pattern (positive slope)

Decreasing at a constant rate:Average rates are the same (negative slope)

Decreasing at an increasing rate: Average rates are a decreasing linear pattern (negative slope)

Pattern remains constant: Average rates are always zero (no change)

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John is growing fickleberries. He plants a fickleberry bush in his backyard. Initially, on the first day the bush is planted, there are 4 fickleberries. Each day, the number of fickleberries increases by 10. Write a recursive equation for this context.

1) We already know what f(1) is: 4.

2) We already know what the formula is: f(n) = f(n-1) + (common difference).

3) So, let's just plug in the values: f(1) = 4, f(n) = f(n-1) + 10

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Need help on summation notation! Click here!

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Sigma notation for sums | Sequences, series and induction | Precalculus | Khan Academy

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To find the geometric means of a sequence, find the (number of missing terms + 1) root of the number you get from dividing the last term by the 1st term.

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To find the arithmetic means of a sequence, subtract the 1st term from the last term and divide it by the number of missing terms (between those terms only!) increased by 1.

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The Shortcut to Finding the Sum of a Sequence

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The Shortcut to Finding the Sum of a Sequence

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There is a shortcut to finding the sum of a sequence (considering if we are given a particular index of terms). Know that the sum of all the terms added up and the product of the average of the 1st and last terms and the number of terms is EQUAL. In other words:

The sum is equal to the average of the 1st and last terms added.

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The formula for writing in summation notation is:

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If dealing with arithmetic sequence:

f(n) = (first term) + (common difference) (n-1)

If dealing with geometric sequence:

f(n) = (first term) x (common ratio) ^n-1

It's a good idea to simplify explicit arithmetic equations!

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f(1) = x
f(n) = f(n-1) +/x (change)

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Recursive Equation - Equation that allows you to find out the NEXT term in a sequence, considering knowing the previous term

Explicit Equation - Equation that allows you to find out ANY term in a sequence

Geometric Sequence = A sequence in which we multiply by a common ratio for each term

Common Ratio - The ratio between two numbers in a geometric sequence AKA the number you multiply terms by in a geometric sequence

Arithmetic Sequence = A sequence in which we add by a common difference for each term

Common Difference - The difference between two numbers in an arithmetic sequence AKA the number you add by in an arithmetic sequence

Summation Notation - Notation for adding up a sequence

Arithmetic Means - Missing terms in an arithmetic sequence

Geometric Means - Missing terms in a geometric sequence

Discrete - Do not connect the dots on a graph; pattern happens only at specific moments

Continuous - Dots are connected on a graph; pattern consistently happens over time