Key Concepts and Formulas

• Geometric Progression (G.P.): 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 ($r$).
• $n$-th term of a G.P.: $a_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.
• Sum of the first $n$ terms of a G.P.: $S_n = \frac{a(r^n - 1)}{r - 1}$ (for $r \neq 1$).

Step-by-Step Solution

Step 1: Define the G.P. and set up equations from the given information. Let the first term of the G.P. be $a$ and the common ratio be $r$. Since the terms are positive, $a > 0$ and $r > 0$. The second, fourth, and sixth terms are $ar$, $ar^3$, and $ar^5$ respectively. Their sum is given as 21: $ar + ar^3 + ar^5 = 21 \quad \text{(Equation 1)}$ The eighth, tenth, and twelfth terms are $ar^7$, $ar^9$, and $ar^{11}$ respectively. Their sum is given as 15309: $ar^7 + ar^9 + ar^{11} = 15309 \quad \text{(Equation 2)}$

Step 2: Simplify the equations by factoring out common terms. Factor out $ar$ from Equation 1: $ar(1 + r^2 + r^4) = 21 \quad \text{(Equation 1 simplified)}$ Factor out $ar^7$ from Equation 2: $ar^7(1 + r^2 + r^4) = 15309 \quad \text{(Equation 2 simplified)}$

Step 3: Solve for the common ratio ($r$) by dividing the simplified equations. Divide Equation 2 simplified by Equation 1 simplified: $\frac{ar^7(1 + r^2 + r^4)}{ar(1 + r^2 + r^4)} = \frac{15309}{21}$ Cancel out common terms ($a$, $r$, and $(1 + r^2 + r^4)$) on the left side: $r^{7-1} = \frac{15309}{21}$ $r^6 = 729$ To find $r$, we need to find the sixth root of 729. We know that $3^6 = 729$. So, $r = 3$ or $r = -3$. Since the G.P. consists of positive terms, the common ratio $r$ must be positive. Therefore, $r = 3$.

Step 4: Solve for the first term ($a$) by substituting the value of $r$ into a simplified equation. Substitute $r=3$ into Equation 1 simplified: $a(3)(1 + 3^2 + 3^4) = 21$ $3a(1 + 9 + 81) = 21$ $3a(91) = 21$ $273a = 21$ $a = \frac{21}{273}$ Simplifying the fraction by dividing numerator and denominator by 21: $a = \frac{1}{13}$

Step 5: Calculate the sum of the first nine terms ($S_9$). We need to find $S_9$ with $a = \frac{1}{13}$, $r = 3$, and $n = 9$. Using the sum formula $S_n = \frac{a(r^n - 1)}{r - 1}$: $S_9 = \frac{\frac{1}{13}(3^9 - 1)}{3 - 1}$ Calculate $3^9$: $3^9 = 19683$. $S_9 = \frac{\frac{1}{13}(19683 - 1)}{2}$ $S_9 = \frac{\frac{1}{13}(19682)}{2}$ $S_9 = \frac{19682}{13 \times 2}$ $S_9 = \frac{19682}{26}$ Performing the division: $S_9 = 757$

Common Mistakes & Tips

• Sign of the Common Ratio: Always check the problem statement for constraints like "positive terms." This is crucial for selecting the correct value of $r$ when solving equations like $r^6 = 729$.
• Factoring Strategy: Recognizing common factors like $(1 + r^2 + r^4)$ is key to simplifying the equations and solving for $r$ efficiently.
• Calculation Accuracy: Be meticulous with calculations involving powers of numbers and fractions to avoid errors in the final answer.

Summary

The problem involves a Geometric Progression with positive terms. By setting up equations based on the given sums of specific terms and then simplifying them through factoring, we were able to solve for the common ratio $r$ and the first term $a$. Finally, the sum of the first nine terms was calculated using the standard G.P. sum formula. The constraint of positive terms was essential in selecting the correct common ratio.

The final answer is $\boxed{757}$.