Sequences & Series
Middle term of AP
Grade 11
Question:
<p>The middle term of the progression \( 2, 5, 8, 11, \ldots, 92 \) is</p>
<p>47</p>
<p>48</p>
<p>46</p>
<p>49</p>
Step-by-Step Solution
Key Concept: Find the number of terms in the AP using the nth term formula aₙ = a + (n-1)d, then identify the middle term using (n+1)/2 for odd n.
<p><strong>Step 1:</strong> Identify AP parameters: a = 2, d = 3 (common difference)</p><p><strong>Step 2:</strong> Find number of terms using aₙ = a + (n-1)d:<br/>92 = 2 + (n-1)×3<br/>90 = 3(n-1)<br/>n = 31 (odd, so middle term exists)</p><p><strong>Step 3:</strong> Find middle term position: Middle position = (31+1)/2 = 16th term</p><p><strong>Step 4:</strong> Calculate 16th term:<br/>a₁₆ = 2 + (16-1)×3 = 2 + 45 = 47</p><p><strong>Verification:</strong> The 16th term divides the 31 terms into 15 terms on each side.</p><p>∴ Answer: <strong>47</strong></p>
Correct Answer: A