Final answer:
The first seven terms of the sequence defined by the recurrence relation T(n) = T(n-1) + 2n-1 with T(1) = 1 are 1, 4, 9, 16, 25, 36, and 49. The sequence represents the series of perfect squares.
Step-by-step explanation:
The student is being asked to calculate the first seven terms of a recursively defined sequence. The recurrence relation provided is T(n) = T(n-1) + 2n-1, with the initial condition T(1) = 1. To find the terms, we repeatedly apply the recurrence to find each term based on the previous one:
- T(1) = 1 (given)
- T(2) = T(1) + 2(2) - 1 = 1 + 3 = 4
- T(3) = T(2) + 2(3) - 1 = 4 + 5 = 9
- T(4) = T(3) + 2(4) - 1 = 9 + 7 = 16
- T(5) = T(4) + 2(5) - 1 = 16 + 9 = 25
- T(6) = T(5) + 2(6) - 1 = 25 + 11 = 36
- T(7) = T(6) + 2(7) - 1 = 36 + 13 = 49
The sequence is a series of perfect squares, with each term given by T(n) = n2. So, the first seven terms are 1, 4, 9, 16, 25, 36, and 49.