Sequences, Series and Imaginary Numbers

Formula to find the value at step n

Tn = a + (n - 1)d

This formula represents the value of nth term of an arithmetic sequence, where a is the first term, n is the term/step number, and d is the common difference.

The question gives you the value of two terms/steps and asks you to find a and d. What do you do?

You sub them both into the Tn formula and subtract them from each other
(a + (n - 1)d) - (a + (n - 1)d)

Find an expression for the nth term Tn

Sub your value for a and d into the Tn formula - leave n in it - and simplify if you can

Find the value of n ….. what do you know n will be?

n will be a whole number (ONLY applies to n!)

This is because n is a step/term in a sequence (you can’t go up 1/3 of a step, you either go up or down that step.)

Find the value of n such that Tn = X OR Tn < X OR Tn > X

1. Use the Tn formula, let it =/</> X.

2. Try to move n to the left of the sign and everything else to the right

3. When you get stuck just sub in values for n until you are =/</> X

4. Make sure to prove your theory by subbing in the value above and below you answer for fiull marks

Sum of first n terms (arithmetic)

Sn = n/2 (a+1)
where a is the first term in the sequence (NOT the previous term)
and n is the term/step you want to sum up to.

Sum of first n terms (geometric)

Sn = a(1 - rⁿ) / (1 - r)
where a is the first term in the sequence (NOT the previous term)
and n is the term/step you want to sum up to.

For an infinite geometric series to exist what must be true?

r must be greater than 0 and less than 1.

Formula to find the nth term in a geometric sequence

Tn = ar^n-1
Where a is the first term, n is the term number and r is the ratio (not infinite so r does not have to be > 0 and < 1)
^{Super important to note here that you MUST do the power on r FIRST - you cannot multiply a by r and then do the power!!}

What’s the difference between a geometric and arithmetic sequence

arithmetic adds a constant each term

geometric multiplies each term by a constant

Whats the conjugate of 2 + 3i

2 - 3i

Whats the conjugate of -2 + 3i

-2 - 3i

Whats the conjugate of 2 - 3i

2 + 3i

In words, how do you solve for 4 + 5i/2-7i

multiply the top and bottom by the conjugate of the bottom -- ( 2 + 7i)

Formula for the modulous

|z| = √a² + b²

Formula for the argument

1. Which quadrant are we in?

2. tan^-1(|b|/|a|)

3. modify based on the quadrant

What do you do if its in quadrant 1 and how do you tell its there?

You do nothing and you can tell its in the 1st quadrant as they are both positive (a +bi)

What do you do if its in quadrant 2 and how do you tell its there?

You add π/2 and you can tell its in the 1st quadrant as the real number is negative and the imaginary is positive (-a + bi)

What is i² equal to?

-1

What is i equal to

√-1