To determine which of the given options is a term in the arithmetic sequence where the 9th term [tex]\( t_9 \)[/tex] is 176 and the 19th term [tex]\( t_{19} \)[/tex] is 376, we can proceed as follows:
### Step 1: Identify the common difference [tex]\( d \)[/tex]
The nth term of an arithmetic sequence can be expressed as:
[tex]\[ t_n = a + (n-1)d \][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
Given:
[tex]\[ t_9 = a + 8d = 176 \][/tex]
[tex]\[ t_{19} = a + 18d = 376 \][/tex]
To find [tex]\( d \)[/tex], subtract the equation for [tex]\( t_9 \)[/tex] from the equation for [tex]\( t_{19} \)[/tex]:
[tex]\[ (a + 18d) - (a + 8d) = 376 - 176 \][/tex]
[tex]\[ 10d = 200 \][/tex]
[tex]\[ d = \frac{200}{10} = 20 \][/tex]
### Step 2: Determine the first term [tex]\( a \)[/tex]
Using the value of [tex]\( d \)[/tex] found, we substitute [tex]\( d \)[/tex] back into the equation for [tex]\( t_9 \)[/tex]:
[tex]\[ t_9 = a + 8d \][/tex]
[tex]\[ 176 = a + 8 \times 20 \][/tex]
[tex]\[ 176 = a + 160 \][/tex]
[tex]\[ a = 176 - 160 = 16 \][/tex]
So, the first term [tex]\( a \)[/tex] is 16.
### Step 3: Find the general term of the sequence
With [tex]\( a = 16 \)[/tex] and [tex]\( d = 20 \)[/tex], the general term [tex]\( t_n \)[/tex] of the sequence is:
[tex]\[ t_n = a + (n-1)d \][/tex]
[tex]\[ t_n = 16 + (n-1) \times 20 \][/tex]
[tex]\[ t_n = 16 + 20n - 20 \][/tex]
[tex]\[ t_n = 20n - 4 \][/tex]
### Step 4: Determine which term is 696
We set up the equation to find [tex]\( n \)[/tex] for [tex]\( t_n = 696 \)[/tex]:
[tex]\[ 20n - 4 = 696 \][/tex]
[tex]\[ 20n = 700 \][/tex]
[tex]\[ n = \frac{700}{20} \][/tex]
[tex]\[ n = 35 \][/tex]
So, 696 is the 35th term of the sequence.
### Conclusion
The only option provided that is a term of the arithmetic sequence is:
a. 696
Thus, the correct answer is a. 696.
