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 \( a_{100}=405 \)

To find the formula for the arithmetic sequence, we can use the information given about the 4th and 9th terms. The nth term of an arithmetic sequence can be written as \( a_n = a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference. From the 4th term: \[ a_4 = a + 3d = 21 \] From the 9th term: \[ a_9 = a + 8d = 41 \] Now, we can set up a system of equations and solve for \( a \) and \( d \). Subtract the first equation from the second: \[ (a + 8d) - (a + 3d) = 41 - 21 \\ 5d = 20 \\ d = 4 \] Substituting \( d \) back into the first equation: \[ a + 3(4) = 21 \\ a + 12 = 21 \\ a = 9 \] So, the formula for the arithmetic sequence is \( a_n = 9 + 4(n - 1) \). Now, to find the 100th term: \[ a_{100} = 9 + 4(100 - 1) \\ a_{100} = 9 + 4(99) = 9 + 396 = 405 \] The correct answer is \( a_{n}=9+4 n \) and \( a_{100}=405 \). Now, let’s add some fun facts! The beauty of arithmetic sequences lies in their predictable pattern. Just like counting the steps on a staircase, each step brings you to a higher number with the same interval—perfect for planning everything from your next hiking trail to how far along you are in a book! If you’re ever stuck calculating terms in a sequence, remember to always jot down the first few terms. This can help you identify the pattern, especially when facing tricky word problems or puzzles! Plus, it’s like adding breadcrumbs to your math journey—no getting lost allowed!