A series has three numbers `a, ar, and ar^2`. In the series, the first term is twice the second term. What is the ratio of the sum of the first two terms to the sum of the last two terms in the series?

Options

A. `1 : 1`
B. `1 : 2`
C. `1 : 4`
D. `2 : 1`

Answer

To solve the problem, we need to analyze the series consisting of three terms: \( a \), \( ar \), and \( ar^2 \). We are given that the first term is twice the second term. Let's break down the solution step by step.

### Step 1: Set Up the Equation
We know from the problem that:
- The first term is \( a \).
- The second term is \( ar \).

According to the problem, the first term is twice the second term:
\[ a = 2(ar) \]

### Step 2: Simplify the Equation
We can simplify the equation from Step 1:
\[ a = 2ar \]
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ 1 = 2r \]
This leads to:
\[ r = \frac{1}{2} \]

### Step 3: Calculate the Sum of the First Two Terms
Now we can calculate the sum of the first two terms:
\[ \text{Sum of the first two terms} = a + ar = a + a \left(\frac{1}{2}\right) = a + \frac{a}{2} = \frac{2a}{2} + \frac{a}{2} = \frac{3a}{2} \]

### Step 4: Calculate the Sum of the Last Two Terms
Next, we calculate the sum of the last two terms:
\[ \text{Sum of the last two terms} = ar + ar^2 = a\left(\frac{1}{2}\right) + a\left(\frac{1}{2}\right)^2 = \frac{a}{2} + \frac{a}{4} \]
To add these fractions, we need a common denominator:
\[ \frac{a}{2} = \frac{2a}{4} \]
Thus,
\[ \text{Sum of the last two terms} = \frac{2a}{4} + \frac{a}{4} = \frac{3a}{4} \]

### Step 5: Find the Ratio of the Sums
Now we find the ratio of the sum of the first two terms to the sum of the last two terms:
\[ \text{Ratio} = \frac{\text{Sum of the first two terms}}{\text{Sum of the last two terms}} = \frac{\frac{3a}{2}}{\frac{3a}{4}} \]
This simplifies to:
\[ = \frac{3a}{2} \cdot \frac{4}{3a} = \frac{4}{2} = 2 \]

### Final Answer
Thus, the ratio of the sum of the first two terms to the sum of the last two terms is:
\[ \text{Ratio} = 2:1 \]