<p>If the <em>p</em>th term of an A.P. is <em>q</em> and the <em>q</em>th term is <em>p</em>, then find its <em>r</em>th term.</p>
Step-by-Step Solution
Key Concept: Use the general term formula aₙ = a + (n-1)d twice to set up equations for the pth and qth terms, then solve for first term a and common difference d to find the rth term.
<p><strong>Step 1:</strong> Let the first term be <em>a</em> and common difference be <em>d</em>.</p><p><strong>Step 2:</strong> Given that <em>p</em>th term = <em>q</em>, so: a + (p-1)d = q ... (1)</p><p><strong>Step 3:</strong> Given that <em>q</em>th term = <em>p</em>, so: a + (q-1)d = p ... (2)</p><p><strong>Step 4:</strong> Subtract equation (2) from equation (1):</p><p>[a + (p-1)d] - [a + (q-1)d] = q - p</p><p>(p-1)d - (q-1)d = q - p</p><p>(p - q)d = q - p</p><p>(p - q)d = -(p - q)</p><p>d = -1</p><p><strong>Step 5:</strong> Substitute d = -1 in equation (1):</p><p>a + (p-1)(-1) = q</p><p>a - p + 1 = q</p><p>a = p + q - 1</p><p><strong>Step 6:</strong> Find the <em>r</em>th term:</p><p>a<sub>r</sub> = a + (r-1)d = (p + q - 1) + (r-1)(-1)</p><p>a<sub>r</sub> = p + q - 1 - r + 1</p><p>∴ <em>a<sub>r</sub></em> = <strong>p + q - r</strong></p>
Correct Answer: p + q - r