Saturday, June 16, 2012

Mathcounts strategies: Some sums

The following sequences are all arithmetic sequences and for any arithmetic sequences, the sum is
always average times the terms (how many numbers).

To find the average, you can use

a. sum divided by how many numbers.
b. average of the first and the last term, the second first and the second last term, or the third first and the third last term, etc...

Sum of the first consecutive natural numbers:

1 + 2 + 3 + 4 + 5 + ... + ( n -1 ) + n = $$\dfrac {n\left( n+1\right) } {2}$$

Examples :

#1: 1 + 2 + 3 + ... + 100 = $$\dfrac {100\left ( 101\right) } {2}=5050$$
#2: 1 + 2 + 3 + ... + 27 = $$\dfrac {27\left( 28\right) } {2}=378$$
#3: 4 + 8 + 12 + ... + 80 = 4 (1 + 2 + 3 + ... + 20) = $$\dfrac {4\times 20\times 21} {2}=840$$

Sum of the first consecutive natural even numbers: Proof without words

Have you noticed in this sequence, every number is double the numbers in the first example,
so you don't need to divide by 2.

$$2+4+6+\ldots +2n=n\left( n+1\right)$$

Examples:

#1: 2 + 4 + 6 + ... 100 = 50 * 51 = 2550
#2: 2 + 4 + 6 + 8 + 10 + 12 = 6 x 7 = 42
In this case, you can also find the midpoint, which is 7 and then 7 x 6 = 42
#3: 2 + 4 + 6 + ...420 = 210 * 211 = 44,310

Sum of the first consecutive odd numbers: Proof without words

In this special case, the mean is the same as how many numbers.
So it's easier if you find the mean by averaging the first and the last term and then square the mean.

1 + 3 + 5 + ... + ( 2n -1) = $$n^{2}$$

Examples:

#1: 1 + 3 + 5 + ... + 39 = $$\left[ \dfrac {\left( 39+1\right) } {2}\right] ^{2}$$ = (20^{2}\)
#2: 1 + 3 + 5 + ... + 89 = $$\left[ \dfrac {89+1} {2}\right] ^{2}=45^{2}=2025$$
#3: 1 + 3 + 5 + ... + 221 = $$\left[ \dfrac {221+1} {2}\right] ^{2}= 111^{2}=12321$$

Applicable problems:

#1: What is the sum of the first 40 consecutive positive integers?
#2: What is the sum of the first 40 consecutive positive even integers?
#3: What is the sum of the first 40 consecutive positive odd integers?
#4: How many times does a 12-hour clock strikes in one day if it strikes once on one o'clock, twice on two o'clock, etc...?
#5: At a game show,you win $100 for the first correct answer and$200 for the second correct answer, etc. How much do you win if you answer 8 questions in a row correctly?

#1: 820
#2: 1640
#3: 1600
#4: 156
#5: 100 + 200 + 300 ... + 800 = 100 (1 + 2 + 3 + ...8) = 100* $$\dfrac {8\times 9} {2}$$ =  3600

Face Diagonal and Space Diagonal of a Rectangular Prism

Face diagonal and space diagonal of a cube

Ways to calculate face and space diagonal.

Each side of the cube is x units long.

Use  45-45-90 degree angle ratio
( 1 - 1 - 2  ) or Pythagorean theorem to get the face diagonal.

Using Pythagorean theorem twice and you'll get the space diagonal.

Face diagonal and Space diagonal of a rectangular prism.

Same way to figure out the face

diagonal of a rectangle as well as

space diagonal of a rectangular prism.

Use Pythagorean theorem or

30-60-90 degree angle ratio

(1 -- 2) to figure out the face

diagonal and Pythagorean theorem

twice to figure out the space diagonal.

Monday, June 11, 2012

Painted Cube Problems: Beginning Level

Please refrain yourself from checking the answers too soon. Use Lego blocks, unit cubes, or Rubik's cubes to help you think. If you are really stuck, check out this link on painted cube problems.

Please comment and help me make my blog more user friendly. Thanks a lot!!
Answer Key to the Painted Cube Problem