The value of $1 + \frac{1}{4 \times 3} + \frac{1}{4 \times 3^2} + \frac{1}{4 \times 3^3}$ is

### Concept & Logic This is a straight arithmetic addition problem involving fractions. Because the denominators contain escalating powers of a base number ($3$), the lowest common multiple (LCM) will naturally be the highest power in the series. ### Step-by-Step Solution First, calculate the actual values of the denominators: $$\text{Term 1} = 1$$ $$\text{Term 2} = \frac{1}{4 \times 3} = \frac{1}{12}$$ $$\text{Term 3} = \frac{1}{4 \times 9} = \frac{1}{36}$$ $$\text{Term 4} = \frac{1}{4 \times 27} = \frac{1}{108}$$ Rewrite the sum with the computed denominators: $$1 + \frac{1}{12} + \frac{1}{36} + \frac{1}{108}$$ Identify the LCM of the denominators. Since $108$ is a multiple of $12$ ($12 \times 9 = 108$) and $36$ ($36 \times 3 = 108$), the common denominator is $108$. Convert all fractions to have $108$ as the denominator: $$1 = \frac{108}{108}$$ $$\frac{1}{12} = \frac{1 \times 9}{12 \times 9} = \frac{9}{108}$$ $$\frac{1}{36} = \frac{1 \times 3}{36 \times 3} = \frac{3}{108}$$ $$\frac{1}{108} = \frac{1}{108}$$ Add the numerators together: $$\text{Sum} = \frac{108 + 9 + 3 + 1}{108}$$ $$= \frac{121}{108}$$ ### Exam Strategy & Shortcut You can estimate the sum to quickly rule out wild options. The sum is $1 + \frac{1}{12} + \text{tiny numbers}$. This is approximately $1.08$. Option (b) is $\frac{3}{2} = 1.5$ (Too big). Option (c) is $\frac{31}{2} = 15.5$ (Far too big). Option (a) is $\frac{121}{108}$, which is roughly $1.12$. This perfectly aligns with our estimation, leaving (a) as the only mathematically plausible option. ### Common Pitfall Some students attempt to use complex geometric progression formulas for the fractional part. While mathematically valid (taking $\frac{1}{12}$ as a common term), it is often slower and more prone to simple arithmetic errors than just converting three small fractions to a common denominator. ### Final Answer **Therefore, the correct answer is 121/108.**