# Algebra 2 – Sequences

Q: What is the tenth term of the sequence: 4, 12, 36, 108, …?

Explanation:  It’s a geometric sequence because each term is multiplied by 3 to get the next term. The common ratio (r) is 3. You *could* just keep multiplying by 3 until you get to the tenth term. It’s not the most efficient way, but it works.

The ‘best’ way, if you know it, is to use the formula for the nth term of a geometric sequence:

Oh, and my TI-84 Plus program ‘NTHTERM’ will find this answer for you.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Geometry – Lines and segments

Q: Points D, E, and F are collinear with DE < EF and DF < EF.  If DE = 2x+6, DF = 3x-4, and EF = 22, what is DE?

Explanation:

First, recall that ‘DE’ refers to the length of the segment with endpoints D and E.  It is the distance between D and E.  Next, DON’T ASSUME that E is between the D and F!  Since DE and DF are each less than EF, and all three points lie on the same line (collinear), D must be between E and F.

Draw the diagram:

__________________________

E                         D                   F

Now label the distances as they are given in the problem:

<————–22——————->

__________________________

E        2x+6         D     3x-4      F

Use the concept: Part + Part = Whole (also called the Segment Addition Postulate) to write the equation:

2x+6 + 3x-4 = 22

Solve:

5x + 2 = 22

5x = 20

x = 4

Plug x=4 back in to the ‘DE’ part:

DE = 2x+6 = 2(4) + 6 = 14.

Tip:  To be sure your answer is correct, plug x=4 into the ‘DF’ part, too:

DF = 3x-4 = 3(4)-4 = 8.  Now, see that the sum of the two ‘parts’ does equal the ‘whole’:

14 + 8 = 22.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Algebra 1, 2 – solving quadratic equations

Q: The square of a positive number is equal to 5 more than the product of 4 and the same number. What is the number?

Explanation:

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – Counting and Ordering

Q: In how many ways can the letters of the word ‘FIVE’ be arranged?

Explanation 1:  SInce there are 4 different letters, do 4!  (that’s ‘4 factorial’).   4! = 4 x 3x 2x 1 = 24.  Your calculator might have a factorial (!) function.  On the TI-84 Plus, find it in MATH – PRB.

Explanation 2:  SInce you are choosing 4 items from a total of 4 items, and the order is important, you can do 4 nPr 4, or use the formula for nPr.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – number properties

Q: How many positive integers less than 500 are multiples of 3, 7, and 11?

Explanation:  3, 7, and 11 do not have any common factors except 1.  So, their least common multiple is 3 x 7 x 11 = 231.  The next two common multiples are 231 x 2 = 462 and 231 x 3 = 693.  693 is greater than 500.  There are only two that are less than 500.  They are 231 and 462.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT: Counting and Ordering

Q: How many ways are there to go from point A to point B on the rectangular prism below, by traveling only along exactly 3 edges of the prism?

Explanation: There are 3 ways to travel away from A, shown below in green. There are 2 ways to continue along each of these paths to B, shown in blue. That’s 3 x 2 = 6 paths to B along three edges of the prism. This is easier than making a mess of the diagram trying to draw all 6 paths (and making it hard to count them).

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – Counting and Ordering

Q:  When 5 people met for dinner, each person shook hands with each other person once.  How many handshakes happened?

Explanation 1: Two people = a handshake.  This question is really ‘How many ways are there to choose 2 people from a total of 5, if the order is not important?’ It’s a combination question. Find the nCr function on your calculator. Type in 5 nCr 2. It’s 10. (There is a longer way to do it by hand, using the formula for nCr.)

Explanation 2: Person A shook hands with 4 people.  That’s  4 handshakes so far. Person B shook hands with 3 other people (his shake with A has already been counted). Person C shook hands with 2 other people.  Person D shook hands with 1 other person (E). Add: 4 + 3 + 2 + 1 = 10.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Algebra 1 – Find the equation of a line

Q:  Find the equation of the line that has the following intercepts:

x-intercept: 5,   y-intercept: -3

Answer:  y = 3/5 x – 3

Explanation:  The x-intercept is the point (5, 0).  The y-intercept is the point (0, -3).  Calculate the slope of the line passing through these points:

m= (0- -3)/(5 – 0) = 3/5

Using slope intercept form (y = mx + b), the equation is y = 3/5 x + b.  We just need to find b.  Or do we?  We could plug in an ordered pair and solve for b, but it’s not necessary.  We already have b, because in slope-intercept form, b always stands for the y-intercept.  So, b = -3.

The equation is: y = 3/5 x – 3

My TI-84 Plus program ‘LINES’ will find this equation for you, given the two intercepts.

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – Remainders

Q: When the positive integer b is divided by 5, the remainder is 1. When b is divided by 7, the remainder is also 1. If b is less than 100, what is the largest possible value for b?

Explanation: b is an integer that is one more than a multiple of both 5 and 7.  The smallest such value is 35 +1 = 36.  But, we need the largest one less than 100.  The next common multiple of 5 and 7 is 70 (do 35 x 2), and the next one is 105 (35 x 3).  Use 70, and add 1.  71.