If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:

Step 1: Formulate equations from given information

Let the first term of the Geometric Progression (G.P.) be $a$ and the common ratio be $r$. The $n$ -th term of a G.P. is given by $a_n = ar^{n-1}$. We are given the sum of the second, fourth, and sixth terms, which forms our first equation. We are also given the sum of the eighth, tenth, and twelfth terms, which forms our second equation.

Step 2: Factorize the equations

We can factor out common terms from both equations. From equation (1), we factor out $ar$. From equation (2), we factor out $ar^7$. This simplification will help us find the common ratio.

Step 3: Find the common ratio 'r'

To find the common ratio $r$, we divide equation (4) by equation (3). This cancels out the common factor $a(1 + r^2 + r^4)$, leaving us with $r^6$. We then solve for $r$ by taking the sixth root of 729. Since the terms are positive, $r$ must be positive.

Step 4: Find the first term 'a'

Now that we have the value of $r$, we can substitute it back into equation (3) to find the first term $a$. We simplify the expression and solve for $a$.

Step 5: Calculate the sum of the first nine terms

The sum of the first $n$ terms of a G.P. is given by the formula $S_n = \frac{a(r^n - 1)}{r - 1}$. We substitute the values of $a = \frac{1}{13}$, $r = 3$, and $n = 9$ into the formula and calculate the sum.

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