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Difference of Arithmetic and Geometric Sequence
(Grade 10 Math)

Arithmetic Sequence - also known as arithmetic progression are sequence which the difference between consecutive terms of a sequence a constant. The constant difference is called COMMON DIFFERENCE.

Illustration:
The sequence 1, 3, 5, 7, .... is Arithmetic Sequence because there have common difference between consecutive terms of sequence. And the common difference of sequence is 2.

How to determine the common difference?
If 7 minus 5 the result is 2 same as if 5 minus 3 and 3 minus 1.

If the ratios of consecutive terms are constant and not equal to zero this sequence called Geometric Sequence. That constant is the common ratio of the sequence.

Illustration:
The sequence 1, 2, 4, 8, .... is Geometric Sequence because there have common ratio between consecutive terms of sequence. And the common ratio of sequence is 2.

How to determine the common ratio?
If 8 divides 4 the result is 2 same as if 4 divides 2 and 2 divides 1.

Basta tandaan mo lang pag Arithmetic Sequence may common DIFFERENCE habang ang Geometric Sequence naman may common RATIO.

Try mo nga! 😁
Identify the following sequence if it is Arithmetic Sequence, Geometric Sequence or not.

1) 3, 9, 27, 81, ...
2) 2, 6, 10, 14, ...
3) 1, 4, 6, 9, 11, ...
4) -5, -7, -9, -11, ...
5) 2, 6, 9, 14, ...
6) 2, 1, 1/2, 1/4, ...

~Sir Miks

Rule of Integers
(Grade 7 Math)

Integers are the natural numbers, along with their negatives and zero.
The set {..., -3, -2, -1, 0, 1, 2, 3, ...} is called the set of integers.

ADDITION of INTEGERS
*To get the sum of two integers with like signs: ADD the addends then copy the common sign.

Illustration:
5 + 6 = 11
-3 + -12 = -15

*To get the sum of two integers with unlike signs: SUBTRACT the addends (highest minus lowest) then copy the sign of the highest number.

Illustration:
1) 5 + (-6) = -1
Between 5 and 6, the highest number is 6 and its sign is negative. That's why the answer is -1.

2) (-3) + 12 = 9
Between 3 and 12, the highest number is 12 and its sign is positive. That's why the answer is positive 9.

SUBTRACTION of INTEGERS
*To subtract an integers, change the subtraction sign ( - ) to addition sign ( + ) and change also the sign of subtrahend. Afterwards used the rule of addition to get the final answer.
Illustration:
1) 5 - 6 = 5 + (-6) = -1
2) (-5) - (-6) = (-5) + 6 = 1
3) (-3) - 7 = (-3) + (-7) = -10
4) 3 - (-7) = 3 + 7 = 10

MULTIPLICATION and DIVISION of INTEGERS
*If the two integers multiply or divides with like sign, their product or quotient is POSITIVE.
Illustration:
1) 5(6) = 30
2) (-2)(-4) = 8
3) 56/8 = 7
4) -60/(-12) = 5

*If the two integers multiply or divides with unlike sign, their product or quotient is NEGATIVE.
Illustration:
1) -5(6) = -30
2) 2(-4) = -8
3) 56/-8 = -7
4) -60/12 = 5

Try mo nga!😁
1. 23 + 34
2. -37 + (-82)
3. 26 - (-74)
4. -56 - (-21)
5. 132/-11
6. 15(-7)

~Sir Miks

Solving the sum and the product of the roots of quadratic equation (ax²+bx+c=0).

To easily compute the sum and the product of the roots of quadratic equation.
Use the formula:
For the sum of roots: -b/a
For the product of roots: c/a

Note: a,b and c are coefficients of quadratic equation and before using the formula make sure na naka general form (ax²+bx+c=0) yung given quadratic equation.

Example 1:
Find the sum and the product of the roots of x²+9x-52=0.

Solution:
Based on the given quadratic equation the value of a,b and c are 1,9 and -52.
By substituting the values to the formula for sum and product of the roots.
The result is:
Sum = -b/a = -(9)/1 = -9
Product = c/a = -52/1 = -52

It means the sum and the product of the roots of quadratic equation x²+9x-52=0 is -9 and -52.

Example 2:
Find the sum and the product of the roots of quadratic equation 2x²-23=10x-11.

Solution:
First, form the given quadratic equation in general form.
2x²-23=10x-11
2x²-23-10x+11=0
2x²-10x-12=0

Now the equation is 2x²-10x-12=0. Based on the equation the value of a,b and c are 2, -10 and -12.

By substituting the values to the formula for sum and product of the roots.
The result is:
Sum = -b/a = -(-10)/2 = 5
Product = c/a = -12/2 = -6

Meaning the sum and the product is 5 and -6.

Try mo nga! 😁
Solve the sum and the product of the roots of the following quadratic equation.
1) x²-5x+6=0
2) 3x²-12x-16=20

~Sir Miks

Solving for the derivative using DELTA METHOD

Note: the symbol like this "∆" read as "Delta".

Problem: What is the derivative of y=2x²+5x+1?

Step1: Substitute x+∆x for x and y+∆y for y:
So y=2x²+5x+1 become y+∆y=2(x+∆x)²+5(x+∆x)+1

Step2: Eliminate y on the left part by subtracting both part by y:
y+∆y=2(x+∆x)²+5(x+∆x)+1
∆y=2(x+∆x)²+5(x+∆x)+1-y

Step3: Substitute the value of y which is 2x²+5x+1 and simplify:
∆y=2(x+∆x)²+5(x+∆x)+1-y
∆y=2(x+∆x)²+5(x+∆x)+1-(2x²+5x+1)
∆y=2(x²+2x∆x+∆x²)+5(x+∆x)+1-2x²-5x-1
∆y=2x²+4x∆x+2∆x²+5x+5∆x+1-2x²-5x-1
∆y=4x∆x+2∆x²+5∆x

Step4: Factor out ∆x from 4x∆x+2∆x²+5∆x then divide both part by ∆x:
∆y=4x∆x+2∆x²+5∆x
∆y=∆x(4x+2∆x+5)

∆y/∆x=∆x(4x+2∆x+5)/∆x
∆y/∆x=4x+2∆x+5

Last Step: Evaluate the limit of ∆y/∆x as ∆x approaches zero ( in other words, substitute zero to ∆x of 4x+2∆x+5):
∆y/∆x=4x+2∆x+5
∆y/∆x=4x+2(0)+5
∆y/∆x=4x+5

It means the derivative of 2x²+5x+1 is 4x+5 😊.

Try mo nga!😁
Solve the derivative of the following using Delta Method.
1) y=x²-100
2) y=2x²+x+18

~Sir Miks