To solve the problem, let's break it down step by step.

Problem Breakdown:

• We are given that the ratio of the first term to the second term is $3:2$.
• The sum of the first three terms of the geometric series is $217.5$.
• We need to find the first term ($a$) and the common ratio ($r$).

Step 1: Express the terms of the geometric series

Let the first term be $a$. Since the ratio of the first term to the second term is $3:2$, we can write the second term as $\frac{2}{3}a$.

The terms of the geometric sequence are:

• First term: $a$
• Second term: $ar$ (where $r$ is the common ratio)
• Third term: $ar^2$

Step 2: Use the ratio of the first two terms

Given that $\frac{a}{ar} = \frac{3}{2}$, we have: $\frac{a}{ar} = \frac{3}{2}$ Simplifying, we find: $r = \frac{2}{3}$

Step 3: Use the sum of the first three terms

The sum of the first three terms is given as $217.5$: $a + ar + ar^2 = 217.5$ Substitute $r = \frac{2}{3}$ into the equation: $a + a\left(\frac{2}{3}\right) + a\left(\frac{2}{3}\right)^2 = 217.5$ Simplify the equation: $a + \frac{2}{3}a + \frac{4}{9}a = 217.5$

Step 4: Combine like terms

Factor $a$ out of the equation: $a \left(1 + \frac{2}{3} + \frac{4}{9}\right) = 217.5$ Convert the fractions to a common denominator: $a \left(\frac{9}{9} + \frac{6}{9} + \frac{4}{9}\right) = 217.5$ $a \left(\frac{19}{9}\right) = 217.5$

Step 5: Solve for $a$

Multiply both sides by $\frac{9}{19}$ to solve for $a$: $a = 217.5 \times \frac{9}{19}$ $a = 103.125$

Step 6: Calculate the common ratio $r$

We already found that $r = \frac{2}{3}$.

Final Answer:

• First term ($a$): $103.125$
• Common ratio ($r$): $\frac{2}{3}$

Would you like further details or have any questions?

Here are five related questions for you:

1. How would the solution change if the ratio of the first term to the second term were $4:5$?
2. What would be the sum of the first four terms of this geometric series?
3. Can you derive the formula for the sum of an infinite geometric series with the given ratio $r$?
4. What happens to the geometric series if the common ratio $r$ is greater than 1?
5. How do you determine if a geometric series converges or diverges?

Tip: Always double-check the calculations, especially when dealing with fractions and their common denominators.