QuestionAvery and Collin were trying to challenge each other with equations for sequences. Avery was looking at an explicit equation that Collin wrote. $$t(n)=4.5 n-8$$ a. Write the first 4 terms for the sequence. b. What would Avery do to write the $15^{\text {th }}$ term of this sequence? c. Write a recursive equation for this sequence.

Studdy Solution
To write a recursive equation, we need to express $t(n)$ in terms of $t(n-1)$. From the explicit formula, the difference between consecutive terms is constant:
$$t(n) - t(n-1) = 4.5n - 8 - (4.5(n-1) - 8)$$ $$t(n) - t(n-1) = 4.5n - 8 - 4.5n + 4.5 - 8$$ $$t(n) - t(n-1) = 4.5$$
Thus, the recursive formula is: $$t(n) = t(n-1) + 4.5$$ with the initial term $t(1) = -3.5$.
The first four terms are: $-3.5, 1, 5.5, 10$. The 15th term is: $59.5$. The recursive equation is: $t(n) = t(n-1) + 4.5$, with $t(1) = -3.5$.