The mean of $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2$ is

### Concept & Formula To find the mean of the squares of the first $n$ natural numbers, we use the standard summation formula divided by $n$. $$ \text{Mean} = \frac{(n+1)(2n+1)}{6} $$ ### Step-by-Step Solution * **Given:** * The series is the squares of the first $7$ natural numbers. * Therefore, $n = 7$. * **Calculation:** * Substitute $n = 7$ into the mean formula. * $\text{Mean} = \frac{(7+1)(2(7)+1)}{6}$ * $\text{Mean} = \frac{8 \times 15}{6}$ * Simplify the expression: * $\text{Mean} = \frac{120}{6} = 20$ ### Exam Strategy & Shortcut Memorizing the direct formula for the *mean* of squares $\frac{(n+1)(2n+1)}{6}$ is much faster than finding the sum of squares using $\frac{n(n+1)(2n+1)}{6}$ and then dividing by $n$. ### Common Pitfall Attempting to calculate the square of each number individually ($1 + 4 + 9 + 16 + 25 + 36 + 49$) and adding them manually. While it works for $n=7$, it becomes incredibly time-consuming and error-prone for larger values of $n$. Use the formula. ### Final Answer **Therefore, the correct answer is 20.**