SUBTRACTION IN FRACTION REGROUPING.pptx

Series Overview
A series is the sum of the terms of a sequence. For example, the series of the first
𝑛
n natural numbers is:

𝑆
𝑛
=
1
+
2
+
3
+
…
+
𝑛
S
n
​
=1+2+3+…+n
Example: Sum of the First
𝑛
n Natural Numbers
The formula for the sum of the first
𝑛
n natural numbers is:

𝑆
𝑛
=
𝑛
(
𝑛
+
1
)
2
S
n
​
=
2
n(n+1)
​

Proof by Mathematical Induction
To prove this formula using mathematical induction, we follow two main steps:

Base Case: Verify the formula for
𝑛
=
1
n=1.

𝑆
1
=
1
=
1
(
1
+
1
)
2
(True)
S
1
​
=1=
2
1(1+1)
​
(True)
Inductive Step: Assume the formula holds for some integer
𝑘
k:

𝑆
𝑘
=
𝑘
(
𝑘
+
1
)
2
S
k
​
=
2
k(k+1)
​

Now we need to prove it for
𝑘
+
1
k+1:

𝑆
𝑘
+
1
=
𝑆
𝑘
+
(
𝑘
+
1
)
S
k+1
​
=S
k
​
+(k+1)
Substituting the inductive hypothesis:

𝑆
𝑘
+
1
=
𝑘
(
𝑘
+
1
)
2
+
(
𝑘
+
1
)
S
k+1
​
=
2
k(k+1)
​
+(k+1)
To combine the terms, we rewrite
(
𝑘
+
1
)
(k+1) with a common denominator:

𝑆
𝑘
+
1
=
𝑘
(
𝑘
+
1
)
2
+
2
(
𝑘
+
1
)
2
=
𝑘
(
𝑘
+
1
)
+
2
(
𝑘
+
1
)
2
S
k+1
​
=
2
k(k+1)
​
+
2
2(k+1)
​
=
2
k(k+1)+2(k+1)
​

Simplifying gives:

𝑆
𝑘
+
1
=
(
𝑘
+
1
)
(
𝑘
+
2
)
2
S
k+1
​
=
2
(k+1)(k+2)
​

This is exactly the formula we want for
𝑛
=
𝑘
+
1
n=k+1. Therefore, by induction, the formula holds for all natural numbers
𝑛
n.

Conclusion
Using this example, we've illustrated how to work with series and apply mathematical induction to prove a formula. If you'd like to explore other types of series or another induction example, let me know!

1. Subtraction of Fraction and Mixed Number

2. OBJECTIVE/S Adds and subtract simple fractions and mixed numbers with and without regrouping.

3. FRACTIONS Is a part of a whole number the number is expressed as a quotient, in which the numerator is divided by the denominator.

4. REGROUPING – regrouping in addition means carrying while regrouping in subtraction means borrowing .

5. STEP 1: Find a common denominator or LCD – Least Common Denominator. In finding denominator we can use the listing method of the number EXAMPLE: 8 – 8, 16, 24 , 32, 40, 48, 56, 64, 72, 80,… 6 – 12, 18, 24 , 30, 36, 42, 48, 54, 60,…. subtract: 6   Steps in subtracting fraction and Mixed Number without regrouping

6. STEP 2: Write equivalent fractions using the common denominator .  

7. STEP 3: Subtract the numerators. The denominator the same. Then add the whole numbers  

8. STEP 4: Simply fractional part if possible

9. STEP 1: Find a common denominator. Multiply the denominators to find their product. EXAMPLE: 6 x 7 = 42 S ubtract: 10   Example 2:Subtracting Fractions and Mixed Numbers with regrouping

10. STEP 2: Write equivalent fractions using the common denominator .  

11.   STEP 3: Subtract the numerators. Copy the denominator. Then subtract the whole numbers if there is any      

12. STEP 4: Rewrite the improper fraction/ get the lowest term  

13. Exercise Directions: Give the difference of the following fractions. 1)   2)   3)   P.S Answer at the back of your LAS show your solution