Select the correct answerThe average of square of natural numbers from 1 to 35 is?Options425526435426

Break Down the Problem

1. Calculate the square of each natural number from 1 to 35.
2. Find the sum of these squares.
3. Divide the sum by the number of natural numbers (which is 35).

Relevant Concepts

• The formula for the sum of the squares of the first $n$ natural numbers is given by: $S = \frac{n(n + 1)(2n + 1)}{6}$

Analysis and Detail

1. Calculate the sum of the squares for $n = 35$:

   • $n = 35$
   • Substitute $n$ into the formula: $S = \frac{35(35 + 1)(2 \cdot 35 + 1)}{6}$
   • First, calculate $(35 + 1) = 36$ and $(2 \cdot 35 + 1) = 71$.
   • Now substitute back into the equation: $S = \frac{35 \cdot 36 \cdot 71}{6}$

2. Perform the multiplication:

   • $35 \cdot 36 = 1260$
   • $1260 \cdot 71 = 89460$

3. Now divide by 6 to get the sum: $S = \frac{89460}{6} = 14910$

4. Calculate the average of the squares:

   • Average = $\frac{S}{n} = \frac{14910}{35}$
   • Calculate this division: $\text{Average} = 426$

Verify and Summarize

• The average of the squares of the first 35 natural numbers is verified and calculated as 426.

Final Answer

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