Question

The mean of the following series is . (rounded off to three decimal places)
\(0.01,\ 0.02,\ 0.03,\ \ldots,\ 1\)

Hint:

This is an AP of 100 terms running from 0.01 to 1; the mean of an AP equals the average of its first and last terms.

Answer 1

Correct Answer: 0.505

Step 1: Identify the series.
The series is \(0.01, 0.02, 0.03, \ldots, 1\). Each term increases by a fixed amount from the one before it, so this is an arithmetic progression (AP) with first term \(a = 0.01\) and common difference \(d = 0.01\).

Step 2: Find the number of terms.
The last term is \(l = 1\). For an AP, the \(n\)-th term is \(a_n = a + (n-1)d\). Setting \(a_n = 1\):
\[ 1 = 0.01 + (n-1)(0.01) \]
\[ 0.99 = (n-1)(0.01) \]
\[ n - 1 = 99 \]
\[ n = 100 \]
So there are 100 terms in the series.

Step 3: Use the mean of an AP.
For any arithmetic progression, the mean equals the average of the first and last terms, since the terms are symmetric about the midpoint.
\[ \text{Mean} = \frac{a + l}{2} = \frac{0.01 + 1}{2} = \frac{1.01}{2} \]

Step 4: Compute the value.
\[ \text{Mean} = 0.505 \]
This value is already exact to three decimal places.

Final Answer:
The mean of the series is \[ \boxed{0.505} \]