Answer :
To solve this problem, we need to find the next common number in the two given sequences after 19.
### Step-by-Step Solution
#### Understanding the Sequences:
1. First Sequence: 7, 11, 15, 19, 23, ...
- Starts at 7.
- Increases by 4 each step.
- General term can be expressed as [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex] where [tex]\( a_1 = 7 \)[/tex] and [tex]\( d = 4 \)[/tex].
- So, [tex]\( a_n = 7 + (n-1) \cdot 4 \)[/tex].
- Simplified: [tex]\( a_n = 4n + 3 \)[/tex].
2. Second Sequence: 1, 10, 19, 28, 37, ...
- Starts at 1.
- Increases by 9 each step.
- General term can be expressed as [tex]\( b_n = b_1 + (n-1) \cdot d \)[/tex] where [tex]\( b_1 = 1 \)[/tex] and [tex]\( d = 9 \)[/tex].
- So, [tex]\( b_n = 1 + (n-1) \cdot 9 \)[/tex].
- Simplified: [tex]\( b_n = 9n - 8 \)[/tex].
#### Finding the Next Common Number:
We need to identify the next number that can be expressed in both forms [tex]\( a_n = 4n + 3 \)[/tex] and [tex]\( b_n = 9m - 8 \)[/tex] for integers [tex]\( n \)[/tex] and [tex]\( m \)[/tex].
Since 19 is a common term in both sequences, start from there:
- For the first sequence, [tex]\( 19 = 4n + 3 \)[/tex]:
- Solving for [tex]\( n \)[/tex]:
[tex]\[ 19 = 4n + 3 \][/tex]
[tex]\[ 19 - 3 = 4n \][/tex]
[tex]\[ 16 = 4n \][/tex]
[tex]\[ n = 4 \][/tex]
- For the second sequence, [tex]\( 19 = 9m - 8 \)[/tex]:
- Solving for [tex]\( m \)[/tex]:
[tex]\[ 19 = 9m - 8 \][/tex]
[tex]\[ 19 + 8 = 9m \][/tex]
[tex]\[ 27 = 9m \][/tex]
[tex]\[ m = 3 \][/tex]
#### Finding the Next Common Term After 19:
To find the next common term:
1. Increment [tex]\( n \)[/tex] and [tex]\( m \)[/tex] from where we left off:
- Current [tex]\( n \)[/tex] and [tex]\( m \)[/tex]: [tex]\( n = 4 \)[/tex] and [tex]\( m = 3 \)[/tex].
2. Calculate the next terms:
- For the first sequence: [tex]\( 4(n+1) + 3 \)[/tex]
- For the second sequence: [tex]\( 9(m+1) - 8 \)[/tex]
Let’s set up an algorithmic approach:
- Continue increasing [tex]\( n \)[/tex] and calculate [tex]\( a_{n+1} \)[/tex].
- Check if [tex]\( a_{n+1} \)[/tex] matches any [tex]\( b_{m} \)[/tex] by increasing [tex]\( m \)[/tex] accordingly.
We can do the calculations iteratively till we find the next term:
First Sequence Next Iterative
[tex]\[ a_5 = 4 \times 5 + 3 = 23 \][/tex]
[tex]\[ a_6 = 4 \times 6 + 3 = 27 \][/tex]
[tex]\[ a_7 = 4 \times 7 + 3 = 31 \][/tex]
[tex]\[ a_8 = 4 \times 8 + 3 = 35 \][/tex]
[tex]\[ a_9 = 4 \times 9 + 3 = 39 \][/tex]
Second Sequence Next Iterative
[tex]\[ b_4 = 9 \times 4 - 8 = 28 \][/tex]
[tex]\[ b_5 = 9 \times 5 - 8 = 37 \][/tex]
[tex]\[ b_6 = 9 \times 6 - 8 = 46 \][/tex]
[tex]\[ b_7 = 9 \times 7 - 8 = 55 \][/tex]
[tex]\[ b_5 = 9 \times 5 - 8 = 37 \][/tex]
Finally, you see 37 matches in both:
So, the next common number after 19 in both sequences is 37.
### Step-by-Step Solution
#### Understanding the Sequences:
1. First Sequence: 7, 11, 15, 19, 23, ...
- Starts at 7.
- Increases by 4 each step.
- General term can be expressed as [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex] where [tex]\( a_1 = 7 \)[/tex] and [tex]\( d = 4 \)[/tex].
- So, [tex]\( a_n = 7 + (n-1) \cdot 4 \)[/tex].
- Simplified: [tex]\( a_n = 4n + 3 \)[/tex].
2. Second Sequence: 1, 10, 19, 28, 37, ...
- Starts at 1.
- Increases by 9 each step.
- General term can be expressed as [tex]\( b_n = b_1 + (n-1) \cdot d \)[/tex] where [tex]\( b_1 = 1 \)[/tex] and [tex]\( d = 9 \)[/tex].
- So, [tex]\( b_n = 1 + (n-1) \cdot 9 \)[/tex].
- Simplified: [tex]\( b_n = 9n - 8 \)[/tex].
#### Finding the Next Common Number:
We need to identify the next number that can be expressed in both forms [tex]\( a_n = 4n + 3 \)[/tex] and [tex]\( b_n = 9m - 8 \)[/tex] for integers [tex]\( n \)[/tex] and [tex]\( m \)[/tex].
Since 19 is a common term in both sequences, start from there:
- For the first sequence, [tex]\( 19 = 4n + 3 \)[/tex]:
- Solving for [tex]\( n \)[/tex]:
[tex]\[ 19 = 4n + 3 \][/tex]
[tex]\[ 19 - 3 = 4n \][/tex]
[tex]\[ 16 = 4n \][/tex]
[tex]\[ n = 4 \][/tex]
- For the second sequence, [tex]\( 19 = 9m - 8 \)[/tex]:
- Solving for [tex]\( m \)[/tex]:
[tex]\[ 19 = 9m - 8 \][/tex]
[tex]\[ 19 + 8 = 9m \][/tex]
[tex]\[ 27 = 9m \][/tex]
[tex]\[ m = 3 \][/tex]
#### Finding the Next Common Term After 19:
To find the next common term:
1. Increment [tex]\( n \)[/tex] and [tex]\( m \)[/tex] from where we left off:
- Current [tex]\( n \)[/tex] and [tex]\( m \)[/tex]: [tex]\( n = 4 \)[/tex] and [tex]\( m = 3 \)[/tex].
2. Calculate the next terms:
- For the first sequence: [tex]\( 4(n+1) + 3 \)[/tex]
- For the second sequence: [tex]\( 9(m+1) - 8 \)[/tex]
Let’s set up an algorithmic approach:
- Continue increasing [tex]\( n \)[/tex] and calculate [tex]\( a_{n+1} \)[/tex].
- Check if [tex]\( a_{n+1} \)[/tex] matches any [tex]\( b_{m} \)[/tex] by increasing [tex]\( m \)[/tex] accordingly.
We can do the calculations iteratively till we find the next term:
First Sequence Next Iterative
[tex]\[ a_5 = 4 \times 5 + 3 = 23 \][/tex]
[tex]\[ a_6 = 4 \times 6 + 3 = 27 \][/tex]
[tex]\[ a_7 = 4 \times 7 + 3 = 31 \][/tex]
[tex]\[ a_8 = 4 \times 8 + 3 = 35 \][/tex]
[tex]\[ a_9 = 4 \times 9 + 3 = 39 \][/tex]
Second Sequence Next Iterative
[tex]\[ b_4 = 9 \times 4 - 8 = 28 \][/tex]
[tex]\[ b_5 = 9 \times 5 - 8 = 37 \][/tex]
[tex]\[ b_6 = 9 \times 6 - 8 = 46 \][/tex]
[tex]\[ b_7 = 9 \times 7 - 8 = 55 \][/tex]
[tex]\[ b_5 = 9 \times 5 - 8 = 37 \][/tex]
Finally, you see 37 matches in both:
So, the next common number after 19 in both sequences is 37.
