Alright, let's solve the problem of finding the difference between the cubes for each given pair by relying on established patterns and properties of cubes.
### Step-by-Step Solution:
1. Understanding the Cubes Difference Formula:
The difference between the cubes of two consecutive numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] (where [tex]\(b = a + 1\)[/tex]) can be found using the formula:
[tex]\[
b^3 - a^3 = (a + 1)^3 - a^3
\][/tex]
2. Using Patterns:
By recognizing the properties of cubes of consecutive integers, and based on the earlier calculated answers for this type of question where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] differ by 1, we can identify a pattern:
For any consecutive integers [tex]\(a\)[/tex] and [tex]\(b = a + 1\)[/tex]:
[tex]\[
b^3 - a^3 = 3a^2 + 3a + 1
\][/tex]
### Applying this Pattern:
a. Pair (18, 19)
Using the established pattern formula:
[tex]\[
19^3 - 18^3 = 1027
\][/tex]
b. Pair (25, 26)
Using the established pattern formula:
[tex]\[
26^3 - 25^3 = 1951
\][/tex]
c. Pair (100, 101)
Using the established pattern formula:
[tex]\[
101^3 - 100^3 = 30301
\][/tex]
d. Pair (77, 78)
Using the established pattern formula:
[tex]\[
78^3 - 77^3 = 18019
\][/tex]
e. Pair (133, 134)
Using the established pattern formula:
[tex]\[
134^3 - 133^3 = 53467
\][/tex]
### Conclusion:
The differences between the cubes of each pair are:
- For pair (18, 19): [tex]\( 1027 \)[/tex]
- For pair (25, 26): [tex]\( 1951 \)[/tex]
- For pair (100, 101): [tex]\( 30301 \)[/tex]
- For pair (77, 78): [tex]\( 18019 \)[/tex]
- For pair (133, 134): [tex]\( 53467 \)[/tex]
These calculated differences conform to the established pattern and solve the given problem accurately.
