To solve the problem, we need to analyze the information given about the geometric progression (G.P.) and use the properties of G.P.s to find the sum of the first two terms. ### Step 1: Understanding the Problem We are told that: - The first term of the G.P. is twice its common ratio. - The sum to infinity of the G.P. is 8. Let’s denote: - The first term as \( a \). - The common ratio as \( r \). From the problem statement, we can express the first term in terms of the common ratio: \[ a = 2r \] ### Step 2: Sum to Infinity of a Geometric Progression The formula for the sum to infinity \( S \) of a G.P. is given by: \[ S = \frac{a}{1 - r} \] This formula is valid only when the absolute value of the common ratio \( |r| < 1 \). Given that the sum to infinity is 8, we can set up the equation: \[ \frac{a}{1 - r} = 8 \] ### Step 3: Substitute the First Term Now, substituting \( a = 2r \) into the sum to infinity formula: \[ \frac{2r}{1 - r} = 8 \] ### Step 4: Solve for \( r \) To solve for \( r \), we can cross-multiply: \[ 2r = 8(1 - r) \] Expanding the right side: \[ 2r = 8 - 8r \] Now, we can combine like terms: \[ 2r + 8r = 8 \] \[ 10r = 8 \] Dividing both sides by 10: \[ r = \frac{8}{10} = \frac{4}{5} \] ### Step 5: Find the First Term \( a \) Now that we have \( r \), we can find \( a \): \[ a = 2r = 2 \times \frac{4}{5} = \frac{8}{5} \] ### Step 6: Calculate the Sum of the First Two Terms The first two terms of the G.P. are: - First term \( a = \frac{8}{5} \) - Second term \( ar = \frac{8}{5} \times \frac{4}{5} = \frac{32}{25} \) Now, we can find the sum of the first two terms: \[ \text{Sum of first two terms} = a + ar = \frac{8}{5} + \frac{32}{25} \] To add these fractions, we need a common denominator. The common denominator between 5 and 25 is 25: \[ \frac{8}{5} = \frac{8 \times 5}{5 \times 5} = \frac{40}{25} \] Now we can add: \[ \frac{40}{25} + \frac{32}{25} = \frac{40 + 32}{25} = \frac{72}{25} \] ### Conclusion: Final Answer Thus, the sum of the first two terms of the G.P. is: **C. \( \frac{72}{25} \)** ### Step 7: Explanation of Other Options - **Option A: \( \frac{8}{5} \)** - This is just the first term, not the sum of the first two terms. - **Option B: \( \frac{8}{3} \)** - This does not correspond to any calculation we performed and is not related to the sum of the first two terms. - **Option D: \( \frac{56}{9} \)** - This value does not match any of our calculations and is incorrect. ### Revision Summary - The first term \( a \) is twice the common ratio \( r \). - The sum to infinity of a G.P. is calculated using \( S = \frac{a}{1 - r} \). - We found \( r = \frac{4}{5} \) and \( a = \frac{8}{5} \). - The sum of the first two terms is \( \frac{72}{25} \).