### Instruction: What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers?### Response: To find the sum of the first 20 positive even integers, I can use a formula for the sum of an arithmetic series.
An arithmetic series is a sequence of numbers with a constant difference between consecutive terms.
The first 20 positive even integers form an arithmetic series with a first term of 2 and a common difference of 2.
The formula for the sum of an arithmetic series is S_n = n/2 (a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
To find a_n, I can use another formula for an arithmetic series: a_n = a_1 + (n - 1) d, where d is the common difference.
Plugging in the values for the even integers, I get a_20 = 2 + (20 - 1) 2 = 2 + 38 = 40.
Therefore, the sum of the first 20 positive even integers is S_20 = 20/2 (2 + 40) = 10 (42) = 420.
Similarly, to find the sum of the first 15 positive odd integers, I can use the same formulas with different values.
The first 15 positive odd integers form an arithmetic series with a first term of 1 and a common difference of 2.
The nth term is a_n = 1 + (n - 1) 2, and the 15th term is a_15 = 1 + (15 - 1) 2 = 1 + 28 = 29.
The sum of the first 15 positive odd integers is S_15 = 15/2 (1 + 29) = 15/2 (30) = 225.
Now I have the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers.
To find the positive difference between them, I need to subtract the smaller sum from the larger sum.
The larger sum is 420 and the smaller sum is 225, so the positive difference is 420 - 225 = 195.