Evaluate the finite series value precisely: 1 + 1/(4 * 3) + 1/(4 * 3^2) + 1/(4 * 3^3). Report the result correct to four decimal places.

Introduction / Context:
This problem asks you to evaluate a short geometric-type expression: 1 + 1/(4 * 3) + 1/(4 * 3^2) + 1/(4 * 3^3). Such questions test your comfort with factoring out constants and using the finite geometric series sum to avoid term-by-term decimal work. The final value must be rounded to four decimal places for a clean numerical answer.

Given Data / Assumptions:

• The expression is 1 + (1/4)*(1/3) + (1/4)*(1/3^2) + (1/4)*(1/3^3).
• Operations follow standard precedence: exponents, then multiplication/division, then addition.
• Round the final result to four decimal places.

Concept / Approach:
Notice that all fractional terms after the initial 1 share a common factor 1/4 and form a geometric series in powers of 1/3. Summing this finite geometric series is faster and more accurate than converting every term to a decimal early. Use the identity for k = 1 to n: sum r^k = r*(1 - r^n)/(1 - r) when |r| < 1.

Step-by-Step Solution:
Write S = 1 + (1/4)[(1/3) + (1/3^2) + (1/3^3)].Sum the geometric part with r = 1/3, n = 3: (1/3)*(1 - (1/3)^3)/(1 - 1/3).Compute: (1/3)*[(1 - 1/27)/(2/3)] = (1/3)*[(26/27)*(3/2)] = (1/3)*(13/9) = 13/27.Multiply by 1/4: (1/4)*(13/27) = 13/108 ≈ 0.120370…Add the leading 1: S ≈ 1.120370… Round to four decimals → 1.1204.

Verification / Alternative check:
Compute each term: 1/(4*3) = 1/12 ≈ 0.083333; 1/(4*9) ≈ 0.027778; 1/(4*27) ≈ 0.009259. Sum of extras ≈ 0.120370; plus 1 gives 1.120370…, confirming the rounded value 1.1204.

Why Other Options Are Wrong:

• 1.1202 / 1.1203 / 1.1201: All are close but result from premature rounding of intermediate steps.
• None of these: Incorrect because a listed option (1.1204) is exact to four decimals.

Common Pitfalls:
Rounding each term too early; forgetting to factor out 1/4; misapplying the finite geometric sum with k starting at 1 instead of 0.

Final Answer:
1.1204