Sequences & Series
Arithmetic Means
Grade 11
Question:
<p>Suppose <math>p</math> is the first of <math>n</math> (<math>n \geq 1</math>) arithmetic means between two positive numbers <math>a</math> and <math>b</math>. The value of <math>p</math> is</p>
<p>(a) <math>\frac{na + b}{n+1}</math></p>
<p>(b) <math>\frac{nb + a}{n+1}</math></p>
<p>(c) <math>\frac{na - b}{n+1}</math></p>
<p>(d) <math>\frac{nb - a}{n+1}</math></p>
Step-by-Step Solution
Key Concept: Arithmetic means divide the interval into equal parts with common difference determined by the total number of terms.
<p>When <math>n</math> arithmetic means are inserted between <math>a</math> and <math>b</math>, there are <math>n+2</math> terms in total. If <math>p</math> is the first arithmetic mean, then <math>a, p, \ldots, b</math> form an AP with common difference <math>d = \frac{b-a}{n+1}</math>. Therefore, <math>p = a + d = a + \frac{b-a}{n+1} = \frac{na + b}{n+1}</math>.</p>
Correct Answer: A