Question

There are exactly four positive integers $n$ such that \[\frac{(n + 1)^2}{n + 23}\] is an integer. Compute the largest such $n$.

Answer 1

Let's present the given equation first. Deciphering the given code, I think the equation is (n+1)²/n+23. Then, we want to find the maximum value of n. Suppose the complete equation is:

f(n) = (n+1)²/n+23

To find the maximum,let's apply the concepts in calculus. The maxima can be determined by setting the first derivative to zero. Therefore, we use the chain rule to differentiate the fraction. For a fraction u/v, the derivative is equal to (vdu-udv)/v².

f'(n) = [(n+23)(2)(n+1)-(n+1)²(1)]/(n+23)² = 0
[(n+23)(2n+2) - (n+1)²]/(n+23)² = 0
(2n²+2n+46n+46-n²-2n-1)/(n+23)²=0
n²+46n+45=0
n = -1, -45

There are two roots for the quadratic equation. Comparing the two, the larger one is -1. Thus, the maximum value of n is -1.