Answer :
To determine which numbers could be terms in the sequence given by [tex]\( a_n = 7n + 4 \)[/tex], we need to check if, for each number, there exists a non-negative integer [tex]\( n \)[/tex] that satisfies the formula.
Let's evaluate each number one by one:
Option A: 46
We set the equation [tex]\( 7n + 4 = 46 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 46 - 4 \][/tex]
[tex]\[ 7n = 42 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{42}{7} \][/tex]
[tex]\[ n = 6 \][/tex]
Since [tex]\( n = 6 \)[/tex] is a non-negative integer, 46 could be a term in the sequence.
Option B: 36
We set the equation [tex]\( 7n + 4 = 36 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 36 - 4 \][/tex]
[tex]\[ 7n = 32 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{32}{7} \][/tex]
[tex]\[ n \approx 4.571 \][/tex]
Since [tex]\( n \approx 4.571 \)[/tex] is not an integer, 36 could not be a term in the sequence.
Option C: 32
We set the equation [tex]\( 7n + 4 = 32 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 32 - 4 \][/tex]
[tex]\[ 7n = 28 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{28}{7} \][/tex]
[tex]\[ n = 4 \][/tex]
Since [tex]\( n = 4 \)[/tex] is a non-negative integer, 32 could be a term in the sequence.
Option D: 24
We set the equation [tex]\( 7n + 4 = 24 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 24 - 4 \][/tex]
[tex]\[ 7n = 20 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{20}{7} \][/tex]
[tex]\[ n \approx 2.857 \][/tex]
Since [tex]\( n \approx 2.857 \)[/tex] is not an integer, 24 could not be a term in the sequence.
Therefore, the numbers that could be terms in the sequence [tex]\( a_n = 7n + 4 \)[/tex] are:
- 46
- 32
So, the correct options are [tex]\(\boxed{\text{A and C}}\)[/tex].
Let's evaluate each number one by one:
Option A: 46
We set the equation [tex]\( 7n + 4 = 46 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 46 - 4 \][/tex]
[tex]\[ 7n = 42 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{42}{7} \][/tex]
[tex]\[ n = 6 \][/tex]
Since [tex]\( n = 6 \)[/tex] is a non-negative integer, 46 could be a term in the sequence.
Option B: 36
We set the equation [tex]\( 7n + 4 = 36 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 36 - 4 \][/tex]
[tex]\[ 7n = 32 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{32}{7} \][/tex]
[tex]\[ n \approx 4.571 \][/tex]
Since [tex]\( n \approx 4.571 \)[/tex] is not an integer, 36 could not be a term in the sequence.
Option C: 32
We set the equation [tex]\( 7n + 4 = 32 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 32 - 4 \][/tex]
[tex]\[ 7n = 28 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{28}{7} \][/tex]
[tex]\[ n = 4 \][/tex]
Since [tex]\( n = 4 \)[/tex] is a non-negative integer, 32 could be a term in the sequence.
Option D: 24
We set the equation [tex]\( 7n + 4 = 24 \)[/tex].
1. Subtract 4 from both sides:
[tex]\[ 7n = 24 - 4 \][/tex]
[tex]\[ 7n = 20 \][/tex]
2. Divide both sides by 7:
[tex]\[ n = \frac{20}{7} \][/tex]
[tex]\[ n \approx 2.857 \][/tex]
Since [tex]\( n \approx 2.857 \)[/tex] is not an integer, 24 could not be a term in the sequence.
Therefore, the numbers that could be terms in the sequence [tex]\( a_n = 7n + 4 \)[/tex] are:
- 46
- 32
So, the correct options are [tex]\(\boxed{\text{A and C}}\)[/tex].
