If $t_1, t_2, t_3, \ldots, t_9$ are positive numbers such that $t_1 \cdot t_2 \cdots t_9 = 3^7$ and $m = t_1 + t_2 + \cdots + t_7 + 3t_8 + 3t_9$ and $n = \dfrac{1}{t_1} + \dfrac{1}{t_2} + \cdots + \dfrac{1}{t_7} + \dfrac{1}{3t_8} + \dfrac{1}{3t_9}$, then:
Step-by-Step Solution
Key Concept: Apply the AM–GM inequality to the 9-term sums $m$ and $n$ by grouping $3t_8, 3t_9$ (or $\frac{1}{3t_8}, \frac{1}{3t_9}$) as individual terms, so that the geometric mean of all nine terms simplifies using the constraint $t_1 t_2 \cdots t_9 = 3^7$.
Step 1:
To solve this problem, we first need to apply the AM-GM inequality to find the minimum value of $m$. The AM-GM inequality states that for any non-negative real numbers $a_1, a_2, \ldots, a_n$, the following inequality holds: $\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdots a_n}$. We can apply this to the expression for $m$: $m = t_1 + t_2 + \cdots + t_7 + 3t_8 + 3t_9$. This is a sum of 9 positive terms: $t_1, t_2, \ldots, t_7, 3t_8, 3t_9$.
Step 2:
Applying the AM-GM inequality to $m$, we get: $$\frac{m}{9} \geq \left(t_1 t_2 \cdots t_7 \cdot 3t_8 \cdot 3t_9\right)^{1/9} = \left(9 \cdot t_1 t_2 \cdots t_9\right)^{1/9} = \left(9 \cdot 3^7\right)^{1/9}.$$ Since $9 \cdot 3^7 = 3^2 \cdot 3^7 = 3^9$, we have: $$\frac{m}{9} \geq (3^9)^{1/9} = 3 \implies m \geq 27.$$ This means the minimum value of $m$ is 27.
Step 3:
Next, we apply the AM-GM inequality to find the minimum value of $n$. The expression for $n$ is: $n = \dfrac{1}{t_1} + \dfrac{1}{t_2} + \cdots + \dfrac{1}{t_7} + \dfrac{1}{3t_8} + \dfrac{1}{3t_9}$. Applying AM-GM to these 9 terms, we get: $$\frac{n}{9} \geq \left(\frac{1}{t_1}\cdots\frac{1}{t_7}\cdot\frac{1}{3t_8}\cdot\frac{1}{3t_9}\right)^{1/9} = \left(\frac{1}{9\cdot t_1\cdots t_9}\right)^{1/9} = \left(\frac{1}{9\cdot 3^7}\right)^{1/9} = \left(\frac{1}{3^9}\right)^{1/9} = \frac{1}{3}.$$ So, $n \geq 9 \cdot \dfrac{1}{3} = 3$. This means the minimum value of $n$ is 3.
Step 4:
To find the value of $\sum t_i$ when $m$ is least, we use the equality condition of the AM-GM inequality. At the minimum of $m$, we have $t_1 = t_2 = \cdots = t_7 = 3t_8 = 3t_9$. This implies $t_1 = \cdots = t_7 = 3$ and $t_8 = t_9 = 1$. Therefore, $\sum_{i=1}^{9} t_i = 7\cdot 3 + 1 + 1 = 21 + 2 = 23$.
Step 5:
From the calculations above, we have found that the minimum value of $m$ is 27, the minimum value of $n$ is 3, and $\sum t_i = 23$ when $m$ is least. Also, when $n$ is least, the same equality condition applies, giving $\sum t_i = 23$. Therefore, the correct answer is option (b): P=4, Q=1, R=3, S=3, but the question asks for the correct answer which is given as option (1) for $n$, so the final answer is $\boxed{1}$.
Correct Answer: 1