The sum of the first n + 2 counting numbers is 43 more than the sum of the first
n counting numbers. What is the value of n?
The sum of the first n counting numbers is given by the formula S1 = (n/2)(n+1).
The sum of the first n + 2 counting numbers is given by the formula S2 = ((n + 2)/2)((n + 2) + 1).
According to the problem, S2 = S1 + 43.
Substituting the formulas for S1 and S2, we get ((n + 2)/2)((n + 2) + 1) = (n/2)(n+1) + 43.
Expanding both sides, we get (n + 2)(n + 3) = n(n + 1) + 86.
Simplifying further, we get n^2 + 5n + 6 = n^2 + n + 86.
By subtracting n^2 + n from both sides, we get 4n + 6 = 86.
Next, we subtract 6 from both sides to get 4n = 80.
Dividing both sides by 4, we get n = 20.
Therefore, the value of n is 20.