Adding Infinite Sequences In The Form: a + a/b + a/b2 + ...

To add infinite sequences in the form: a + ^a/_b + ^a/_b2 + ...

Use the formula a/(1-r) where a is the first number, or the numerator, and r is whatever a is being multiplied, or 1/b.

EX: 2 + 2/3 + 2/9 + 2/37 + ...

= 2/(1- (1/3)) = 2/(2/3) = 3

EX: 4 + 2 + 1 + ...

= 4/(1- (1/2)) = 4/(1/2) = 8

-Marisa(:

Cubic Roots

There's a simple way to find the cubed root of a number.

First, look at the last digit of the number. If the number ends in...

0, the last digit ends in 0.
1, the last digit ends in 1.
2, the last digit ends in 8.
3, the last digit ends in 7.
4, the last digit ends in 4.
5, the last digit ends in 5.
6, the last digit ends in 6.
7, the last digit ends in 3.
8, the last digit ends in 2.
9, the last digit ends in 9.

Once you determine the last digit, mentally chop off the last three digits of the number under the cube root sign. Then, use your knowledge of cubic numbers to find out what cubic numbers the remaining digits are between. Write the smaller of these two numbers.

Ex: The cubed root of 373248  = __________.

1) The last digit is 8, so the new number will end in 2.
2) "Chopping off" the last three digits of the number leaves us with 373. What two cubed numbers does this fall between?
3)  7 cubed is 343 and 8 cubed is 512, so the first digit is 7. Write down the 7.
4) Therefore, the answer is 72.

Ex 2: The cubed root of 57392
1) The last digit is 2, so the new number will end in 8.
2) "Chopping off" the last three digits of the number leaves us with 57.
3) 3 cubed is 27, 4 squared is 64. Therefore, the first number is 3.
4) Thus, the number is 38. (*Note: The true cubic root of 57392 is 38.573. This approximates a close answer that will suffice on the Number Sense test for certain problems.)

-Marisa

Which is Larger?

Nearly every Number Sense test has a problem asking which is smaller, and offers you two fractions. Though this might seem simple, I always get mixed up when they provide negative fractions, so I decided to dedicate a blog to this subject.

If the two fractions are positive, it is easy to determine which is smaller and which is larger. Simply use the zipper method!

EX: 3/4 amd 2/3, which is larger?

Using the zipper method, 3*3 = 9, and 4*2 = 8. 9 is greater than 8, and in the zipper method would be found under 3/4. Thus, 3/4 is larger than 2/3.

With negative numbers, the larger number is the number closer to zero. Though in the above example 3/4 was larger to 2/3, if both numbers were negative, -2/3 would be LARGER than -3/4, because it is closer in range to 0.

-Marisa

Subtracting Special Fractions

This trick requires for the fractions to be in the special form
X (X+1)/(X+2) - Y (Y+1)(Y+2)

An example of this is 6 7/8 - 1 2/3.

The answer X-Y (X-Y)/(denominators multiplied together) so the solution to the exapmle above is
(6-1) (6-1)/(8*3) = 5 5/24

This applies to both positive and negative answers

Example: 2 3/4 - 6 7/8
(2-6) (2-6)(4*8) = -4 4/34 = -4 1/8 REMEMBER TO REDUCE!

-Kevin

Approximating square roots

On questions that are multiples of ten, you are able to approximate, and some of these include a square root of a relatively large number. You can easily approximate this number by taking off sets of two numbers from the end and approximating the first 3-5 numbers, then add a 0 for every 2 you took off.

Ex: √(139456)

take off the last 2, so the 56, and you're left with 1394. 35² is 1225, and 40² is 1600, and that 1394 is about in the middle, so about 37². Write it down, then add the zero from taking 2 off the end.

370 is what you would write down.

Ex: √(8675309)

if you take off the last 4 digits, you are left with 867. 25² = 625, and 30² is 900. 867 is closer to 30².
Then you would write down 29, and then add 2 zeroes (for the 4 numbers we took off).
So write down 2900.

Subtracting Reverses

There are two forms of this trick.
1. 3-digit reverses(753 - 357)
Subtract the 1st digit of the 1st number and the 1st digit of the 2nd number
7 - 3 = 4
Then plug your answer into the equation: 100n - n to get the final answer.
100(4) - 4 = 396
753 - 357 = 396
2. 4-digit reverse pairs(7568 - 6875)
Subtract the 1st pair from the second pair.
75 - 68 = 7
Then plug your answer into the equation: 100n-n to get the final answer.
100(7) - 7 = 693
7568 - 6875 = 693

Adding a Sequence in the Form 1 + 3 + .... + 2n - 1

When you have a sequence in the format similar to:

1 + 3 + 5 + 7 + ... + 2n - 1

1) Take the last number presented and add it to 1.

2) Divide the number by 2.

3) Square the number.

You now have the total of the sequence!

Ex: 1 + 3 + 5 + ... + 27

1) 27 + 1 = 28

2) 28/2 = 14

3) 14² = 196

Therefore, 1 + 3 + 5 + ... + 27 = 196

Ex: 1 + 3 + 5 + ... + 55

1) 55 + 1 = 56

2) 56/2 = 28

3) 28² = 784

Now try:
Ex: 1 + 3 + 5 + ... + 15

Ex: 1 + 3 + 5 + ... + 39

Ex: 1 + 3 + 5 + ... + 31