If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1, then the value of f(2023) is ______.

The product of the roots of the equation \(\log_2 2^{(\log_2 x)^2} - 5\log_2 x + 6 = 0\) is ______.

If \(f(n) = 1 + 2 + 3 + \cdots + (n+1)\) and \(g(n) = \sum_{k=1}^{n} \frac{1}{f(k)}\), then the least value of n for which g(n) exceeds the value \(\frac{99}{100}\) is ______.

The total number of positive integer solutions of \(21 \le a + b + c \le 25\) is ______.

The polynomial \(4x^{10} - x^9 + 3x^8 - 5x^7 + cx^6 + 2x^5 - x^4 + x^3 - 4x^2 + 6x - 2\) when divided by x - 1 leaves a remainder 2. Then the value of c + 6 is ______.

If three consecutive coefficients in the expansion of \((x + y)^n\) are in the ratio 1:9:63, then the value of n is ______.

The remainder when \(1! + 2! + 3! + \cdots + 95!\) is divided by 15 is ______.

Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km journey, if Vinita travels equal di…

Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum possible value of a + c is ______.