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# Arithmetic Progressions

What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.

What is the arithmetic mean of \( 3a +15 \) and \( 7a -9 \)?

\[ 8, x, 16, 18 \]

If the average of the above numbers is 13, what is \(x \)?

What is the arithmetic mean of all the prime numbers less than 20?

What is the arithmetic mean of all the leap years in the \(19^{\text{th}}\) century?

**Details and Assumptions:**

A year that is divisible by 4 is a leap year. However, a year that is divisible by 100 but not by 400 is a common year.

The \(19^{\text{th}}\) begins with January 1, 1801 and ends with December 31, 1900.

\[ -10, -6, -2, 2, 6, 10 \] What is the arithmetic mean of the above sequence?

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