Problem Analysis:
The worksheet involves identifying patterns in number sequences and determining specific terms in those sequences. Each sequence follows a consistent arithmetic progression, where each term increases by a fixed difference from the previous term.
Solution Approach:
1.
Identify the common difference in each sequence.
2.
Use the formula for the nth term of an arithmetic sequence:
\[
a_n = a_1 + (n-1) \cdot d
\]
where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
Detailed Solutions:
####
Problem 1: Sequence: 25, 35, 45, 55
-
First term (\( a_1 \)): 25
-
Common difference (\( d \)): \( 35 - 25 = 10 \)
##### a. Tenth number (\( n = 10 \)):
\[
a_{10} = 25 + (10-1) \cdot 10 = 25 + 9 \cdot 10 = 25 + 90 = 115
\]
##### b. Twentieth number (\( n = 20 \)):
\[
a_{20} = 25 + (20-1) \cdot 10 = 25 + 19 \cdot 10 = 25 + 190 = 215
\]
##### c. Hundredth number (\( n = 100 \)):
\[
a_{100} = 25 + (100-1) \cdot 10 = 25 + 99 \cdot 10 = 25 + 990 = 1015
\]
####
Problem 2: Sequence: 100, 200, 300, 400
-
First term (\( a_1 \)): 100
-
Common difference (\( d \)): \( 200 - 100 = 100 \)
##### a. Tenth number (\( n = 10 \)):
\[
a_{10} = 100 + (10-1) \cdot 100 = 100 + 9 \cdot 100 = 100 + 900 = 1000
\]
##### b. Twentieth number (\( n = 20 \)):
\[
a_{20} = 100 + (20-1) \cdot 100 = 100 + 19 \cdot 100 = 100 + 1900 = 2000
\]
##### c. Hundredth number (\( n = 100 \)):
\[
a_{100} = 100 + (100-1) \cdot 100 = 100 + 99 \cdot 100 = 100 + 9900 = 10000
\]
####
Problem 3: Sequence: 60, 80, 100, 120
-
First term (\( a_1 \)): 60
-
Common difference (\( d \)): \( 80 - 60 = 20 \)
##### a. Tenth number (\( n = 10 \)):
\[
a_{10} = 60 + (10-1) \cdot 20 = 60 + 9 \cdot 20 = 60 + 180 = 240
\]
##### b. Twentieth number (\( n = 20 \)):
\[
a_{20} = 60 + (20-1) \cdot 20 = 60 + 19 \cdot 20 = 60 + 380 = 440
\]
##### c. Hundredth number (\( n = 100 \)):
\[
a_{100} = 60 + (100-1) \cdot 20 = 60 + 99 \cdot 20 = 60 + 1980 = 2040
\]
####
Problem 4: Sequence: 1000, 2000, 3000, 4000
-
First term (\( a_1 \)): 1000
-
Common difference (\( d \)): \( 2000 - 1000 = 1000 \)
##### a. Tenth number (\( n = 10 \)):
\[
a_{10} = 1000 + (10-1) \cdot 1000 = 1000 + 9 \cdot 1000 = 1000 + 9000 = 10000
\]
##### b. Twentieth number (\( n = 20 \)):
\[
a_{20} = 1000 + (20-1) \cdot 1000 = 1000 + 19 \cdot 1000 = 1000 + 19000 = 20000
\]
##### c. Hundredth number (\( n = 100 \)):
\[
a_{100} = 1000 + (100-1) \cdot 1000 = 1000 + 99 \cdot 1000 = 1000 + 99000 = 100000
\]
Final Answers:
1.
- a. \( \boxed{115} \)
- b. \( \boxed{215} \)
- c. \( \boxed{1015} \)
2.
- a. \( \boxed{1000} \)
- b. \( \boxed{2000} \)
- c. \( \boxed{10000} \)
3.
- a. \( \boxed{240} \)
- b. \( \boxed{440} \)
- c. \( \boxed{2040} \)
4.
- a. \( \boxed{10000} \)
- b. \( \boxed{20000} \)
- c. \( \boxed{100000} \)
Parent Tip: Review the logic above to help your child master the concept of math patterns worksheet grade 4.
