Write a formula for an arithmetic sequence where the 4 th term is 21 and the 9 th term is 41 . Then, use the formula to determine the value of the 100th term. $$ a_{n}=5+4 n $$ and $$ a_{100}=405 $$ $$ a_{n}=9+4 n $$ and $$ a_{100}=409 $$ $$ a_{n}=5+4 n $$ and $$ a_{100}=409 $$ $$ a_{n}=9+4 n $$ and $$ a_{100}=405 $$

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To find the formula for the arithmetic sequence where the 4th term is 21 and the 9th term is 41, we can use the general formula for the $$ n $$-th term of an arithmetic sequence:

$$
a_n = a_1 + (n-1)d
$$

where $$ a_1 $$ is the first term and $$ d $$ is the common difference.

### Step 1: Set up the equations

From the information given:
- The 4th term $$ a_4 = 21 $$:
$$
a_1 + 3d = 21 \quad \text{(1)}
$$

- The 9th term $$ a_9 = 41 $$:
$$
a_1 + 8d = 41 \quad \text{(2)}
$$

### Step 2: Solve the system of equations

We can subtract equation (1) from equation (2) to eliminate $$ a_1 $$:

$$
(a_1 + 8d) - (a_1 + 3d) = 41 - 21
$$
$$
5d = 20
$$
$$
d = 4
$$

Now, substitute $$ d = 4 $$ back into equation (1) to find $$ a_1 $$:

$$
a_1 + 3(4) = 21
$$
$$
a_1 + 12 = 21
$$
$$
a_1 = 9
$$

### Step 3: Write the formula for the sequence

Now that we have $$ a_1 = 9 $$ and $$ d = 4 $$, we can write the formula for the $$ n $$-th term:

$$
a_n = 9 + (n-1) \cdot 4
$$
$$
a_n = 9 + 4n - 4
$$
$$
a_n = 4n + 5
$$

### Step 4: Calculate the 100th term

Now, we can find the 100th term $$ a_{100} $$:

$$
a_{100} = 4(100) + 5
$$
$$
a_{100} = 400 + 5
$$
$$
a_{100} = 405
$$

### Conclusion

The formula for the arithmetic sequence is:

$$
a_n = 4n + 5
$$

And the value of the 100th term is:

$$
a_{100} = 405
$$

Thus, the correct option is:

$$ a_{n}=5+4 n $$ and $$ a_{100}=405 $$
The formula for the arithmetic sequence is $$ a_n = 4n + 5 $$, and the 100th term is 405.

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