The first even number is 2. So, the sum of the first 1 even number is 2.
The first two even numbers are 2 and 4. So, the sum of the first 2 even numbers is 2 + 4 = 6.
The first three even numbers are 2, 4, and 6. So, the sum of the first 3 even numbers is 2 + 4 + 6 = 12.
The first four even numbers are 2, 4, 6, and 8. So, the sum of the first 4 even numbers is 2 + 4 + 6 + 8 = 20.
From these observations, we can notice a pattern:
For n = 1, the sum is 2.
For n = 2, the sum is 6.
For n = 3, the sum is 12.
For n = 4, the sum is 20.
It appears that the sum of the first n even numbers can be represented as a function of n. Let's denote this sum as ∑n. Using inductive reasoning, we can derive a formula based on the pattern we've observed:
∑n = 2 + 4 + 6 + ... + (2n)
To find a formula, we can factor out the common term of 2:
∑n = 2(1 + 2 + 3 + ... + n)
Now, we need to find a formula for the sum of the first n positive integers, which is a well-known arithmetic series formula:
Sum of the first n positive integers = n(n + 1))/2
So, substituting this into our equation:
∑n = 2*(n(n + 1)/2)
Now, simplify:
∑n = n(n + 1)
Therefore, the formula for the sum of the first n positive even integers using inductive reasoning is:
∑n = n(n + 1)