Sequences & Series
Arithmetic Progression - General Term
Grade 11
Question:
<p>Find the first negative term of the sequence <span>\(20, 19\frac{3}{4}, 19\frac{1}{2}, ...\)</span></p>
Step-by-Step Solution
Key Concept: For an AP with negative common difference, find when the general term becomes negative by solving the inequality t_n < 0.
<p><strong>Step 1:</strong> Identify the AP parameters. First term <span>$a = 20$</span>, common difference <span>$d = -\frac{3}{4}$</span></p><p><strong>Step 2:</strong> Set the nth term less than zero: <span>$t_n < 0$</span></p><p><strong>Step 3:</strong> <span>$a + (n-1)d < 0$</span></p><p><span>$20 + (n-1)\left(-\frac{3}{4}\right) < 0$</span></p><p><span>$80 - 3n + 3 < 0$</span></p><p><span>$83 < 3n$</span></p><p><span>$n > 27\frac{2}{3}$</span></p><p><strong>Step 4:</strong> The smallest integer value is <span>$n = 28$</span></p><p>∴ The 28th term is the first negative term.</p>
Correct Answer: 28