We are asked to find the common ratio $r$ and the first term $a$ of a geometric sequence where the sum of the first two terms is 15 and the sum to infinity is 27.

Step 1: Use the formula for the sum to infinity

The sum to infinity $S_\infty$ of a geometric series is given by: $S_\infty = \frac{a}{1 - r}$ We are told that the sum to infinity is 27, so: $\frac{a}{1 - r} = 27$ This is equation (1).

Step 2: Use the sum of the first two terms

The sum of the first two terms of a geometric sequence is: $a + ar = 15$ Factor out $a$: $a(1 + r) = 15$ This is equation (2).

Step 3: Solve the system of equations

From equation (1):

$a = 27(1 - r)$ Substitute this into equation (2): $27(1 - r)(1 + r) = 15$ Simplify the expression: $27(1 - r^2) = 15$ Divide both sides by 27: $1 - r^2 = \frac{15}{27} = \frac{5}{9}$ $r^2 = 1 - \frac{5}{9} = \frac{4}{9}$ $r = \frac{2}{3}$ Since all terms are positive, $r = \frac{2}{3}$.

Step 4: Find $a$

Substitute $r = \frac{2}{3}$ into equation (1): $\frac{a}{1 - \frac{2}{3}} = 27$ $\frac{a}{\frac{1}{3}} = 27$ $a = 27 \times \frac{1}{3} = 9$

Final Answer:

• (i) The common ratio $r$ is $\frac{2}{3}$.
• (ii) The first term $a$ is 9.

Would you like further details or clarifications on any step?

Related Questions:

1. What is the sum of the first three terms of the sequence?
2. How would the common ratio affect the sequence if it were greater than 1?
3. Can you determine the 5th term of this geometric sequence?
4. How would the sum to infinity change if the first term doubled?
5. What is the formula for the $n$-th term in a geometric sequence?

Tip: In a geometric sequence, the sum to infinity exists only when the common ratio $r$ is between -1 and 1 (exclusive).