The sequence of number `a, ar, ar^2 and ar^3` are in geometric progression. The sum of the first four terms in the series is 5 times the sum of first two terms and `r != -1` and `a != 0`. How many times larger is the fourth term than the second term?

Options

A. 1
B. 2
C. 4
D. 5

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Identify the Terms
The sequence given is \( a, ar, ar^2, ar^3 \), where:
- \( a \) is the first term,
- \( r \) is the common ratio.

### Step 2: Write the Sum of the Terms
The sum of the first four terms is:
\[ S_4 = a + ar + ar^2 + ar^3 = a(1 + r + r^2 + r^3) \]
Using the formula for the sum of a geometric series, we can express this as:
\[ S_4 = a \frac{r^4 - 1}{r - 1} \]

The sum of the first two terms is:
\[ S_2 = a + ar = a(1 + r) \]
This can also be expressed as:
\[ S_2 = a \frac{r^2 - 1}{r - 1} \]

### Step 3: Set Up the Equation
According to the problem, the sum of the first four terms is 5 times the sum of the first two terms:
\[ S_4 = 5S_2 \]
Substituting the expressions for \( S_4 \) and \( S_2 \):
\[ a \frac{r^4 - 1}{r - 1} = 5 \left( a \frac{r^2 - 1}{r - 1} \right) \]

### Step 4: Simplify the Equation
Since \( a \neq 0 \) and \( r - 1 \neq 0 \) (as \( r \neq -1 \)), we can cancel \( a \) and \( r - 1 \) from both sides:
\[ r^4 - 1 = 5(r^2 - 1) \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ r^4 - 1 = 5r^2 - 5 \]
\[ r^4 - 5r^2 + 4 = 0 \]

### Step 6: Let \( x = r^2 \)
Let \( x = r^2 \). The equation becomes:
\[ x^2 - 5x + 4 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 - 16}}{2} \]
\[ x = \frac{5 \pm 3}{2} \]
Thus, we have two solutions:
\[ x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{2}{2} = 1 \]
So, \( r^2 = 4 \) or \( r^2 = 1 \).

### Step 8: Find \( r \)
Since \( r^2 = 4 \), we have:
\[ r = 2 \quad \text{or} \quad r = -2 \]
Since \( r^2 = 1 \), we have:
\[ r = 1 \quad \text{or} \quad r = -1 \]
However, \( r \neq -1 \) as per the problem statement.

### Step 9: Calculate the Ratio of the Fourth Term to the Second Term
We need to find how many times larger the fourth term \( ar^3 \) is than the second term \( ar \):
\[ \frac{ar^3}{ar} = r^2 \]
Since we found \( r^2 = 4 \), we conclude:
\[ \frac{ar^3}{ar} = 4 \]

### Conclusion
Thus, the fourth term is 4 times larger than the second term.