98th term of the infinite series $1, 2, 3, 4, 1, 2, 3, 4, 1, 2, \dots$ is

### Concept & Formula To find a specific term in a periodic, repeating sequence, use modular arithmetic. Divide the target term number by the length of the repeating cycle to find the remainder, which points to the exact position within the base cycle. ### Step-by-Step Solution **Given:** * The infinite sequence is $1, 2, 3, 4, 1, 2, 3, 4 \dots$ * Target position $= 98$th term. **Calculation / Deduction:** Identify the repeating block (the cycle) of the series. The sequence repeats the block "$1, 2, 3, 4$". The length of this cycle is $4$ terms. To find the 98th term, divide 98 by the cycle length (4): $98 \div 4 = 24$ with a remainder of $2$. This mathematical result means the full block of "$1, 2, 3, 4$" repeats exactly $24$ times (accounting for $96$ terms). The 97th term begins a new cycle (starting at $1$). The 98th term is the 2nd number in the cycle. Looking at the base cycle "$1, 2, 3, 4$", the 2nd number is $2$. ### Exam Strategy & Shortcut Use the divisibility rule for 4 (look at the last two digits, which is just 98 here). The closest multiple of 4 below 98 is 96. Simply subtract: $98 - 96 = 2$. A remainder of 2 always means the answer is the 2nd term in the repeating sequence. ### Common Pitfall A common mistake is confusing the quotient with the remainder. A student might calculate $98 \div 4 = 24.5$ and blindly guess "4" because of the 4 in the quotient, or they might calculate a remainder of 0 if their division is flawed, leading them to pick the last term of the cycle. ### Final Answer Therefore, the correct answer is **2**.