Solution: The sequence increases by 3 each time. The nth term is $2 + (n-1) \times 3 = 3n - 1$. Set $3n - 1 = 32$: $3n = 33$, so $n = 11$. $\boxed{11}$### Question 1
["Understanding Linear Sequences: Find the 11th Term Using a Simple Formula", "When exploring sequences in mathematics, one of the most fundamental skills is identifying patterns and expressing them mathematically. A classic example is an arithmetic sequence, where each term increases by a constant difference.", "In this sequence, we observe that the first term is 2, and every subsequent term increases by 3. This is a perfect arithmetic progression with a common difference of 3.", "### Formula for the nth Term\nThe general formula for the nth term of an arithmetic sequence is:\n[\na_n = a_1 + (n - 1) \cdot d\n]\nWhere:\n- (a_n) = nth term\n- (a_1) = first term\n- (d) = common difference\n- (n) = term number", "Plugging in the values from our sequence:\n(a_1 = 2), (d = 3)\n[\na_n = 2 + (n - 1) \cdot 3\n]\nSimplify the expression:\n[\na_n = 2 + 3n - 3 = 3n - 1\n]\nSo, the nth term is given by:\n[\na_n = 3n - 1\n]", "### Finding the 11th Term\nTo find the 11th term ((a_{11})), substitute (n = 11) into the formula:\n[\na_{11} = 3(11) - 1 = 33 - 1 = 32\n]\nThis confirms that when (n = 11), the term is 32 — exactly what the sequence predicts.", "### Why This Formula Matters\nUnderstanding and applying such formulas helps solve problems efficiently, whether in algebra, data modeling, or algorithm design. Recognizing sequences that increase by a constant value streamlines the process of identifying terms without listing every element.", "### Final Answer\nThe solution to the problem — determining that the 11th term is 32 — comes directly from solving:\n[\n3n - 1 = 32 \Rightarrow 3n = 33 \Rightarrow n = 11\n]\nSo, the value of (n) is:\n[\n\boxed{11}\n]", "---", "This clear method applies to any arithmetic sequence and strengthens foundational math intuition. Next time you spot a regular pattern increasing by 3, remember the simple formula to unlock the exact term!"]
