Difficulty: Medium
Correct Answer: 90.222222222
Explanation:
Introduction / Context:
This question involves a decimal series where each term has an increasing number of zeros after the decimal point before the digit 2. It is a nice example of applying geometric series ideas to decimal numbers. Instead of trying to add all terms directly, which can be messy, we can recognise a pattern and rewrite the sum in a more convenient algebraic form. Such problems test the ability to work with place values and geometric progressions in decimal form.
Given Data / Assumptions:
Concept / Approach:
Each term can be viewed as 10 plus a fractional part. If we separate the constant 10 from each term, we can sum the integer contributions and the decimal contributions independently. The decimal parts form a geometric progression with first term 0.2 and common ratio 0.1. Using the sum formula for a geometric progression, we can compute the total of the decimal parts exactly. Then, adding the result to the sum of the integer parts gives the final answer.
Step-by-Step Solution:
Step 1: Observe that each term is 10 plus a decimal part: 10.2 = 10 + 0.2, 10.02 = 10 + 0.02, 10.002 = 10 + 0.002, and so on.Step 2: There are 9 terms, so the integer part contributes 9 * 10 = 90.Step 3: The decimal parts form the series 0.2 + 0.02 + 0.002 + 0.0002 + 0.00002 + 0.000002 + 0.0000002 + 0.00000002 + 0.000000002.Step 4: This decimal series is a geometric progression with first term a = 0.2 and common ratio r = 0.1, since each term is obtained by multiplying the previous term by 0.1.Step 5: The sum of the first n terms of a geometric progression is S = a * (1 - r^n) / (1 - r).Step 6: Here, n = 9, a = 0.2, and r = 0.1, so S = 0.2 * (1 - 0.1^9) / (1 - 0.1).Step 7: Compute the denominator: 1 - 0.1 = 0.9.Step 8: The numerator is 0.2 * (1 - 0.1^9). Since 0.1^9 is very small (0.000000001), we keep it symbolically to stay exact: S = 0.2 * (1 - 0.000000001) / 0.9.Step 9: So S = 0.2 / 0.9 * (1 - 0.000000001) = (2 / 9) * (0.999999999).Step 10: The exact value is 0.222222222, because (2 / 9) * (0.999999999) equals 0.222222222.Step 11: Therefore, the total sum of the original series is 90 (integer part) plus 0.222222222 (decimal part) = 90.222222222.
Verification / Alternative check:
We can approximate by ignoring the extremely tiny term 0.000000002 temporarily. Add the first few decimal parts: 0.2 + 0.02 + 0.002 + 0.0002 + 0.00002 ≈ 0.22222. Adding the rest brings it close to 0.222222222. When this is added to 90, the result is about 90.2222, which is consistent with the exact calculation of 90.222222222. Thus the chosen option matches both the exact and approximate reasoning.
Why Other Options Are Wrong:
Option 90.022222222: This is too small, as it effectively reduces the contribution from the decimal series.Option 92.222222222: This assumes the integer part sums to something larger than 90, which is impossible as we only have nine tens.Option 89.222222222: This would result from using 9 * 9 instead of 9 * 10 for the integer part.Option 90.202222222: This slightly underestimates the geometric sum of the decimal series and does not match the exact calculation.
Common Pitfalls:
A frequent mistake is to try to add all decimals manually without recognising the geometric pattern, leading to rounding errors. Another error is to forget that each term includes 10, and instead only add the decimal parts. Some learners also misapply the geometric series formula or mix up the values of a and r. Keeping the structure clear, splitting the sum into integer and decimal parts, and then methodically applying the geometric sum formula avoids these issues.
Final Answer:
The value of 10.2 + 10.02 + 10.002 + ... up to 9 terms is 90.222222222.
Discussion & Comments