Select the correct answerThe average of square of natural numbers from 1 to 35 is?Options426425526435

1. Break Down the Problem

To find the average of the squares of natural numbers from 1 to 35, we will follow these steps:

1. Calculate the squares of each natural number from 1 to 35.
2. Sum all the squared values.
3. Divide the total sum by the number of natural numbers (which is 35).

2. Relevant Concepts

The formula for the average is given by: $\text{Average} = \frac{\text{Sum of Squared Values}}{\text{Total Count of Values}}$

3. Analysis and Detail

1. Calculate the squares: The squares of natural numbers from 1 to 35 can be represented as: $1^2, 2^2, 3^2, \ldots, 35^2$

2. Sum of squares: The sum of squares formula for the first $n$ natural numbers is: $S = \frac{n(n + 1)(2n + 1)}{6}$ For $n = 35$: $S = \frac{35(35 + 1)(2 \cdot 35 + 1)}{6}$ $S = \frac{35 \cdot 36 \cdot 71}{6}$

   • Calculating: $35 \cdot 36 = 1260$ $1260 \cdot 71 = 89460$ $S = \frac{89460}{6} = 14910$

3. Average calculation: Thus, the average is: $\text{Average} = \frac{14910}{35} = 426$

4. Verify and Summarize

We have verified the computations, and the sum of squares of the first 35 natural numbers is 14910. Dividing this sum by 35 gives us the average:

Final Answer

The average of the squares of natural numbers from 1 to 35 is 426.