Final answer:
To determine the nᵗʰ term of an arithmetic sequence where the third term is 32 and the fifth term is 46, find the common difference, determine the first term, and use the formula for the nᵗʰ term, resulting in 18 + (n-1)(7).
Step-by-step explanation:
The problem involves determining the nᵗʰ term of an arithmetic sequence given that the third term is 32 and the fifth term is 46. In an arithmetic sequence, the difference between consecutive terms (common difference) is constant.
To find the nᵗʰ term, we first find the common difference and then use the formula for the nᵗʰ term, which is an = a1 + (n-1)d, where a1 is the first term and d is the common difference.
Here's how to find it step-by-step:
- First, find the common difference (d) by subtracting the third term from the fifth term and then dividing by the difference in their positions (2 since 5 - 3 = 2).
- Once we have d, we can find the first term (a1) by using one of the given terms and solving the formula for a1.
- Finally, insert a1 and d into the formula for the nth term to determine the expression for any term in the sequence.
Let's solve this:
- d = (46 - 32) / (5 - 3) = 14 / 2 = 7
- To find the first term, we can use the third term (32) and the common difference (7): 32 = a1 + 2(7), so a1 = 32 - 14 = 18.
- The nth term, an, is then given by an = 18 + (n-1)(7).