This problem requires us to find the sum of squares of 100 terms, given their mean and standard deviation. We will use the relationship between mean, standard deviation, and the sum of squares.

Understanding the Given Information

• Number of terms, n = 100
• Mean of the terms, $\bar{x}$ = 50
• Standard deviation of the terms, $\sigma$ = 3

Relevant Statistical Formulas

The formula for the mean ($\bar{x}$) is given by:

$$ \bar{x} = \frac{\sum x}{n} $$

The formula for the variance ($\sigma^2$) is given by:

$$ \sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2 $$

From this variance formula, we can rearrange it to solve for the sum of squares ($\sum x^2$):

$$ \frac{\sum x^2}{n} = \sigma^2 + (\bar{x})^2 $$ $$ \sum x^2 = n \left( \sigma^2 + (\bar{x})^2 \right) $$

Step-by-Step Calculation

1. Calculate the variance: Since the standard deviation ($\sigma$) is 3, the variance ($\sigma^2$) is $3^2$. $$ \sigma^2 = 3^2 = 9 $$
2. Calculate the square of the mean: The mean ($\bar{x}$) is 50, so the square of the mean is $50^2$. $$ (\bar{x})^2 = 50^2 = 2500 $$
3. Calculate the sum of squares ($\sum x^2$): Using the rearranged variance formula: $$ \sum x^2 = n \left( \sigma^2 + (\bar{x})^2 \right) $$ Substitute the values we have: n = 100, $\sigma^2$ = 9, and $(\bar{x})^2$ = 2500. $$ \sum x^2 = 100 (9 + 2500) $$ $$ \sum x^2 = 100 (2509) $$ $$ \sum x^2 = 250900 $$

Conclusion

The sum of squares of the 100 terms is 2,50,900.