1. State the problem: Given the sequence defined by $a_1 = 1$ and the recurrence relation $a_{n+1} = a_n + n + 1$, find a closed-form expression for $a_n$ and compute $a_{100}$. 2. Understand the recurrence: The term $a_{n+1}$ is obtained by adding $n + 1$ to the previous term $a_n$. 3. Write out the first few terms to identify a pattern: - $a_1 = 1$ - $a_2 = a_1 + 1 + 1 = 1 + 2 = 3$ - $a_3 = a_2 + 2 + 1 = 3 + 3 = 6$ - $a_4 = a_3 + 3 + 1 = 6 + 4 = 10$ - $a_5 = a_4 + 4 + 1 = 10 + 5 = 15$ 4. Recognize the pattern of $a_n$: The sequence 1, 3, 6, 10, 15,... corresponds to the triangular numbers which are given by the formula $$a_n = \frac{n(n+1)}{2}.$$ 5. Verify the formula satisfies the recurrence: - Base case: $a_1 = \frac{1 \times 2}{2} = 1$ which matches. - Assume $a_n = \frac{n(n+1)}{2}$ - Then: $$a_{n+1} = a_n + n + 1 = \frac{n(n+1)}{2} + n + 1 = \frac{n(n+1) + 2(n+1)}{2} = \frac{(n+1)(n+2)}{2}$$ - This matches the formula with $n$ replaced by $n+1$, so the formula is correct. 6. Compute $a_{100}$: $$a_{100} = \frac{100 \times 101}{2} = 50 \times 101 = 5050.$$