Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>If <i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>, <i>A</i><sub>3</sub>, ..., <i>A</i><sub>m</sub> are arithmetic means between −3 and 828, and the sum of these arithmetic means equals 14025, find the value of <i>m</i>.</p>
Step-by-Step Solution
Key Concept: The sum of m arithmetic means between two numbers equals m times the arithmetic mean of those two numbers.
<p><strong>Step 1:</strong> The sum of arithmetic means between two numbers <i>a</i> and <i>b</i> is given by: $$A_1 + A_2 + ... + A_m = m\left(\frac{a + b}{2}\right)$$</p><p><strong>Step 2:</strong> Substituting <i>a</i> = −3 and <i>b</i> = 828: $$A_1 + A_2 + ... + A_m = m\left(\frac{-3 + 828}{2}\right) = m\left(\frac{825}{2}\right)$$</p><p><strong>Step 3:</strong> Given that the sum equals 14025: $$m \cdot \frac{825}{2} = 14025$$</p><p><strong>Step 4:</strong> Solving for <i>m</i>: $$m = \frac{14025 \times 2}{825} = \frac{28050}{825} = 34$$</p><p>∴ <i>m</i> = 34</p>
Correct Answer: 34