1. **Stating the problem:** We are given a sequence with terms corresponding to positions $n=100, 101, 102$ having values $412, 416, 420$ respectively. We want to find the general formula for the $n^{th}$ term of this sequence. 2. **Formula and rules:** For an arithmetic sequence, the $n^{th}$ term is given by: $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term and $d$ is the common difference. 3. **Find the common difference $d$:** Calculate the difference between consecutive terms: $$d = 416 - 412 = 4$$ Check with the next term: $$420 - 416 = 4$$ So, the sequence increases by 4 each time. 4. **Find the first term $a_1$:** We know the term at position $100$ is $412$, so: $$a_{100} = a_1 + (100-1)d = a_1 + 99d = 412$$ Substitute $d=4$: $$a_1 + 99 \times 4 = 412$$ $$a_1 + 396 = 412$$ $$a_1 = 412 - 396 = 16$$ 5. **Write the general formula:** $$a_n = 16 + (n-1) \times 4 = 16 + 4n - 4 = 4n + 12$$ 6. **Interpretation:** The $n^{th}$ term of the sequence is $a_n = 4n + 12$. This means for any position $n$, multiply $n$ by 4 and add 12 to get the term. **Final answer:** $$a_n = 4n + 12$$