Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>When the ninth term of an AP is divided by its second term, we get 5 as the quotient. When the thirteenth term is divided by the sixth term, the quotient is 2 and the remainder is 5. Find the second term of the AP.</p>
Step-by-Step Solution
Key Concept: Convert the division conditions into linear equations using the division algorithm (dividend = divisor × quotient + remainder), then solve the system of equations.
<p><strong>Step 1:</strong> Let <mathjax>a</mathjax> be the first term and <mathjax>d</mathjax> be the common difference.</p><p><strong>Step 2:</strong> From the first condition: <mathjax>T_9 = 5T_2</mathjax></p><p><mathjax>(a + 8d) = 5(a + d)</mathjax></p><p><mathjax>a + 8d = 5a + 5d</mathjax></p><p><mathjax>3d = 4a</mathjax> ... (i)</p><p><strong>Step 3:</strong> From the second condition: <mathjax>T_{13} = T_6 \cdot 2 + 5</mathjax></p><p><mathjax>a + 12d = 2(a + 5d) + 5</mathjax></p><p><mathjax>a + 12d = 2a + 10d + 5</mathjax></p><p><mathjax>2d = a + 5</mathjax> ... (ii)</p><p><strong>Step 4:</strong> From equation (i): <mathjax>a = \frac{3d}{4}</mathjax></p><p><strong>Step 5:</strong> Substitute in equation (ii): <mathjax>2d = \frac{3d}{4} + 5</mathjax></p><p><mathjax>8d = 3d + 20</mathjax></p><p><mathjax>5d = 20</mathjax></p><p><mathjax>d = 4</mathjax></p><p><strong>Step 6:</strong> From (i): <mathjax>a = 3</mathjax></p><p><strong>Step 7:</strong> <mathjax>T_2 = a + d = 3 + 4 = 7</mathjax></p><p>∴ The second term is <strong>7</strong>.</p>
Correct Answer: 7