$ ext{If the sum of } n ext{ terms of a geometric progression is } 5(2^n - 1), ext{ then common ratio is :}$

Q: $\text{If the sum of } n \text{ terms of a geometric progression is } 5(2^n - 1), \text{ then common ratio is :}$

$\text{If the sum of } n \text{ terms of a geometric progression is } 5(2^n - 1), \text{ then common ratio is :}$

Question

$	ext{If the sum of } n 	ext{ terms of a geometric progression is } 5(2^n - 1), 	ext{ then common ratio is :}$

Accepted Answer

To determine the common ratio of the geometric progression (GP) given that the sum of $ n $ terms is $ 5(2^n - 1) $, we need to analyze the formula for the sum of a GP.

The sum $ S_n $ of the first $ n $ terms of a geometric progression with the first term $ a $ and common ratio $ r $ (where $ r \neq 1 $) is given by:

$$
S_n = a \frac{r^n - 1}{r - 1}
$$

According to the problem, the sum of the first $ n $ terms is $ 5(2^n - 1) $. Therefore, we equate the two expressions for the sum:

$$
a \frac{r^n - 1}{r - 1} = 5(2^n - 1)
$$

To find the common ratio $ r $, let's assume the first term $ a = 5 $. Substituting $ a = 5 $ into the equation, we have:

$$
5 \frac{r^n - 1}{r - 1} = 5(2^n - 1)
$$

Dividing both sides by 5, we get:

$$
\frac{r^n - 1}{r - 1} = 2^n - 1
$$

For this equation to hold true for all $ n $, the expressions on both sides must be equivalent. This implies that:

$$
r^n - 1 = (r - 1)(2^n - 1)
$$

Expanding the right-hand side, we have:

$$
r^n - 1 = r \cdot 2^n - 2^n - r + 1
$$

For the equation to be valid for all $ n $, the coefficients of $ 2^n $ on both sides must match, which implies $ r = 2 $.

Thus, the common ratio $ r $ is 2, which corresponds to Option 2.

**Explanation for Incorrect Options:**

1. **Option 1: $\frac{1}{2}$**

If $ r = \frac{1}{2} $, substituting into the sum formula would not satisfy the equation $ \frac{r^n - 1}{r - 1} = 2^n - 1 $ for all $ n $. The left-hand side would not simplify to match the right-hand side.

2. **Option 3: $\frac{1}{3}$**

Similarly, if $ r = \frac{1}{3} $, the sum formula would not equate to $ 2^n - 1 $ for all $ n $. The expressions would not be equivalent, leading to a mismatch.

3. **Option 4: 3**

If $ r = 3 $, substituting into the sum formula would result in a left-hand side that grows faster than $ 2^n - 1 $, thus not satisfying the equation for all $ n $.

Therefore, the only value of $ r $ that satisfies the given condition for all $ n $ is 2, making Option 2 the correct answer.