Key Concepts and Formulas

• Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The $n$-th term is given by $a_n = a_1 r^{n-1}$, where $a_1$ is the first term.
• Properties of GP terms: For a GP of positive increasing numbers, $a_1 > 0$ and $r > 1$.
• Algebraic Manipulation: Simplifying expressions by factoring and substitution.

Step-by-Step Solution

Step 1: Understand the Problem and Define Variables We are given a Geometric Progression (GP) of increasing positive numbers, denoted by $a_1, a_2, a_3, \dots$. This means the first term $a_1 > 0$ and the common ratio $r > 1$. The general term of a GP is $a_n = a_1 r^{n-1}$. We are provided with two conditions:

1. The product of the fourth and sixth terms is 9: $a_4 a_6 = 9$.
2. The sum of the fifth and seventh terms is 24: $a_5 + a_7 = 24$. We need to find the value of the expression $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$.

Step 2: Translate the Given Conditions into Equations Using the formula $a_n = a_1 r^{n-1}$:

• Condition 1: $a_4 = a_1 r^3$ and $a_6 = a_1 r^5$. So, $a_4 a_6 = (a_1 r^3)(a_1 r^5) = a_1^2 r^8 = 9$. Since $a_1 > 0$ and $r > 1$, $a_1 r^4$ must be positive. Taking the square root of both sides of $a_1^2 r^8 = 9$, we get $|a_1 r^4| = 3$, which simplifies to $a_1 r^4 = 3$. Let's call this equation $(*)$.

• Condition 2: $a_5 = a_1 r^4$ and $a_7 = a_1 r^6$. So, $a_5 + a_7 = a_1 r^4 + a_1 r^6 = 24$.

Step 3: Solve for Intermediate Values using the Equations We have the equation from Condition 2: $a_1 r^4 + a_1 r^6 = 24$. We can factor out $a_1 r^4$ from the left side: $a_1 r^4 (1 + r^2) = 24$. From equation $(*)$, we know that $a_1 r^4 = 3$. Substituting this value into the factored equation: $3 (1 + r^2) = 24$. Divide by 3: $1 + r^2 = 8$. Subtract 1: $r^2 = 7$. Since $r > 1$, we have $r = \sqrt{7}$. Now, we can find $a_1$ using equation $(*)$: $a_1 r^4 = 3$. Since $r^2 = 7$, $r^4 = (r^2)^2 = 7^2 = 49$. So, $a_1 (49) = 3$, which means $a_1 = \frac{3}{49}$.

Step 4: Evaluate the Target Expression The expression to evaluate is $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$. Let's evaluate each part:

• $a_1 a_9$: $a_1 a_9 = a_1 (a_1 r^8) = a_1^2 r^8 = (a_1 r^4)^2$. Using $a_1 r^4 = 3$, we get $(3)^2 = 9$.

• $a_2 a_4 a_9$: $a_2 a_4 a_9 = (a_1 r)(a_1 r^3)(a_1 r^8) = a_1^3 r^{1+3+8} = a_1^3 r^{12}$. This can be written as $(a_1 r^4)^3$. Using $a_1 r^4 = 3$, we get $(3)^3 = 27$.

• $a_5 + a_7$: This sum is directly given in the problem as 24.

Now, substitute these values back into the target expression: $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7 = 9 + 27 + 24$. $9 + 27 + 24 = 36 + 24 = 60$.

Common Mistakes & Tips

• Sign Ambiguity: When taking square roots (e.g., $\sqrt{9}$), remember that the "increasing positive numbers" condition ($a_1 > 0, r > 1$) is crucial for selecting the correct positive root for terms like $a_1 r^4$.
• Simplify Before Substituting: Instead of finding explicit values for $a_1$ and $r$ and substituting them everywhere, look for common intermediate expressions like $a_1 r^4$. This significantly reduces calculations and potential errors.
• Utilize Given Information Directly: The sum $a_5 + a_7 = 24$ is given. Do not recalculate it from $a_1$ and $r$ at the final step; use the given value directly.

Summary The problem requires understanding the properties of a Geometric Progression, particularly how to express terms using the first term and common ratio. By translating the given conditions into algebraic equations, we derived a key intermediate result ($a_1 r^4 = 3$). This intermediate result, along with the direct information given ($a_5 + a_7 = 24$), allowed for a straightforward evaluation of the target expression by simplifying each term. The conditions of the GP being of increasing positive numbers were essential for resolving any sign ambiguities.

The final answer is $\boxed{60}$.