nth term in a sequence

Geometric Sequence








e.g. Find 7th term of (2, 6, 18, 54, ...)
n = 7;  a₁ = 2, r = 3 An = 1458



 Arithmetic Sequence 

e.g. Find 10th term of (3, 5, 7, 9, ...)

n = 10;  a₁ = 3, d = 2 An = 21

Sum of Geometric Series

Sum of Finite Geometric Series

Sum of Infinite Geometric Series

Combination Probability

Dependent Probability

KA has not given a lot of exercises to master probabilities. This scenario keeps coming up so I am posting as reminder of the process /reasoning behind dependent probabilities.

Note to self: associate it with the formulas:
P(A) x P(B|A) = P(B)
P(A and B) = P (A) x P (B)
P(B) x P (A|B) = P(A)

Graphing Trig Functions (Horizontal Phase Shift)

I have been learning Trigonometry in the KhanAcademy site in the past weeks as a sort of a hobby. I have been drawing in previous months and now I am interested in relearning math. This started when I was helping a cousin with his home studies when I realized how much of my highschool math I have forgotten. I was also not surprised to find out that we have not covered everything in highschool due to the slow progression of topics having to repeat the lessons for class mates who had difficulties understanding.  Anyway, I am now on the topic of Graphing Trig Functions. I have struggled with horizontal phase shifts so I am here noting down what has helped me understand it better so incase I forget again and I have to go back to the lessons, I can refer to this as a cheat sheet.

Problem: A pendulum is swinging next to a wall. The distance from the bob of the pendulum to the wall has period 0.8 seconds, amplitude 6 cm, and midline y=15 cm. At time t=0.5, the bob is at its midline, moving toward the wall.

Answer can be a plus horizontal shift or a minus horizontal shift.

Tip discovered to answer these questions:
1. Where is the peak (for cosine) or average (for sine), call it "t" - distance in the x line, this can be negative or positive
2. You want your sine /Cosine argument to equal to 0 or 2pi, so what do you need to do to "t"? add or subtract?

image below:

e.g.
The distance to the peak is positive .5seconds, so I need to remove .5 seconds
i.e. t-0.5 OR t +0.3


This is a fresh discovery after having struggled answering quizzes in the past 3 days so it is not as clear as I want it to be but it makes sense for me so for now that is all that matters. I will redraft when I can.

Consecutive Numbers

I derived these shortcuts in high school. I'm noting it down because I always forget the formula and I have to derive it again to arrive to it.


Finding the sum(y) of x consecutive numbers (n).
xn + (x^2-x)/2 = y
x(n + (x-1)/2 = y

Finding the sum(y) of x consecutive EVEN/ODD numbers (n)
xn + x2 - x = y
x(n+x-1) = y