1. **Problem statement:** Find the 50th term of the sequence 5, 8, 13, 20, ... 2. **Identify the pattern:** The sequence is not arithmetic (difference between terms is not constant) nor geometric (ratio between terms is not constant). Let's check the differences: $$8 - 5 = 3$$ $$13 - 8 = 5$$ $$20 - 13 = 7$$ The differences are 3, 5, 7, which increase by 2 each time. 3. **Define the difference sequence:** Let the first difference be $d_1 = 3$, and the difference increases by 2 each time, so the $n$th difference is: $$d_n = 3 + (n-1) \times 2 = 2n + 1$$ 4. **Express the $n$th term:** The $n$th term $a_n$ can be found by summing the first term and all previous differences: $$a_n = a_1 + \sum_{k=1}^{n-1} d_k = 5 + \sum_{k=1}^{n-1} (2k + 1)$$ 5. **Calculate the sum:** $$\sum_{k=1}^{n-1} (2k + 1) = 2 \sum_{k=1}^{n-1} k + \sum_{k=1}^{n-1} 1 = 2 \frac{(n-1)n}{2} + (n-1) = (n-1)n + (n-1) = (n-1)(n+1) = n^2 - 1$$ 6. **Substitute back:** $$a_n = 5 + n^2 - 1 = n^2 + 4$$ 7. **Find the 50th term:** $$a_{50} = 50^2 + 4 = 2500 + 4 = 2504$$ **Final answer:** The 50th term of the sequence is **2504**.