Let A = (1 / 0.4) + (1 / 0.04) + (1 / 0.004) + ... up to 8 terms of this pattern. Each term is obtained by dividing 1 by a decimal that is one tenth of the previous denominator. What is the exact value of A?

Introduction / Context:
This question involves a geometric progression built from reciprocals of decimal numbers. Each term is formed by dividing 1 by a power of 0.1 multiplied by 0.4, so the denominators get ten times smaller each step. Recognizing the geometric pattern allows us to sum the series efficiently using the standard formula for a finite geometric progression, rather than computing each term separately in an ad hoc way.

Given Data / Assumptions:

• A = (1 / 0.4) + (1 / 0.04) + (1 / 0.004) + ... for 8 terms total.
• Each new denominator is one tenth of the previous denominator.
• All terms are positive real numbers.
• We are asked for the exact sum of these 8 terms.

Concept / Approach:
We first compute the first term and the common ratio. Let the first term be a and the common ratio be r. Then the sum of n terms of a geometric progression is: S_n = a (r^n - 1) / (r - 1), for r ≠ 1. In this problem, each denominator is reduced by a factor of 10, so each term is multiplied by 10 relative to the previous term, meaning that the series is a geometric progression with ratio r = 10. We then apply the formula for n = 8.

Step-by-Step Solution:
Step 1: Compute the first term a = 1 / 0.4. Since 0.4 = 4 / 10, 1 / 0.4 = 10 / 4 = 2.5, so a = 2.5. Step 2: Find the second term 1 / 0.04 = 25, which is 10 times bigger than 2.5. Step 3: Therefore, the common ratio is r = 10. Step 4: There are n = 8 terms in total. Step 5: Use the finite geometric sum formula: A = a (r^8 - 1) / (r - 1) = 2.5 (10^8 - 1) / (10 - 1). Step 6: Compute 10^8 = 100000000, so 10^8 - 1 = 99999999. Step 7: Divide by 9: 99999999 / 9 = 11111111. Step 8: Multiply by 2.5: A = 2.5 * 11111111 = 27777777.5.

Verification / Alternative check:
To verify, list all eight terms explicitly: 2.5, 25, 250, 2500, 25000, 250000, 2500000 and 25000000, then add them carefully or with a calculator. The sum is 27777777.5, which matches the result obtained using the geometric series formula. This confirms that the pattern was correctly identified and the formula was applied without mistakes.

Why Other Options Are Wrong:
The other options 27272727.5, 25252525.5, 25555555.5 and 26543210.5 are close in magnitude but do not match the exact value from the geometric series calculation. They represent likely outcomes if someone miscounts the number of terms, miscomputes the common ratio or makes a slip in division by 9. Only 27777777.5 corresponds exactly to the correct sum of the eight terms.

Common Pitfalls:
Students sometimes misinterpret the notation and think the series is 0.4 + 0.04 + 0.004, which is a completely different problem. Another frequent error is to assume the ratio is 0.1 rather than recognizing that the terms themselves grow by a factor of 10 because we are taking reciprocals of smaller and smaller decimals. Mistakes in computing large powers of 10 or dividing 99999999 by 9 can also lead to wrong answers. Writing the pattern explicitly for the first few terms is a good way to avoid these confusions.

Final Answer:
Thus, the exact value of A is 27777777.5.