Sequences & Series
Geometric Progression
Grade 11
Question:
<p>Let <i>a</i> be the first term and <i>r</i> be the common ratio of a GP. If <i>ar</i><sup>2</sup> and <i>ar</i><sup>4</sup> are two terms of the GP whose product is 25, and <i>ar</i> + <i>ar</i><sup>3</sup> = <span>\(\frac{25}{2}\)</span>, find the sum of three consecutive terms of the form <i>ar</i><sup>3</sup>, <i>ar</i><sup>5</sup>, <i>ar</i><sup>7</sup>.</p>
Step-by-Step Solution
Key Concept: Use the given conditions on the GP terms to find the first term and common ratio, then compute the required sum.
Step 1: From the given condition, $ar^2 \cdot ar^4 = 25$.
$$a^2r^6 = 25$$
Taking the positive square root, we have:
$$ar^3 = 5$$
Step 2: The common ratio is $r=2$.
Step 3: Using $ar^3 = 5$ and $r=2$:
$$a(2^3) = 5$$
$$8a = 5$$
$$a = \frac{5}{8}$$
Step 4: The terms to be summed are $ar^3, ar^5, ar^7$. These terms are $4, 8, 16$.
Step 5: The sum of these terms is:
$$4 + 8 + 16 = 28$$
Correct Answer: 28