Let be a continuous function defined on the interval satisfying the following condition:for .Define a sequence by for all natural numbers . Find the natural number such thatoindent
[THOUGHT] The integral from a_n to a_{n+1} simplifies to (5/3) k^2 - k where k = 2^{n-1}. Setting this equal to N/3 with N = 1407374833221632 gives 5k^2 - 3k = N. Solve the quadratic 5p^2 - 3p - N = 0 for p = [3 + sqrt(9 + 20N)] / 10, check if it's an integer power of 2, then n = log2(p) + 1. [/THOUGHT] [PYTHON] import math
N = 1407374833221632 D = 9 + 20 * N s = int(math.sqrt(D) + 0.5) is_square = (s * s == D) p_num = 3 + s if p_num % 10 == 0: p = p_num // 10 is_power_of_two = (p > 0) and (p & (p - 1) == 0) if is_power_of_two: exponent = p.bit_length() - 1 n = exponent + 1 print(n) else: print("Not power of 2") else: print("Not divisible by 10")
print("D:", D) print("sqrt approx:", s) print("is square:", is_square) print("p:", p_num // 10 if p_num % 10 == 0 else "not int") [/PYTHON]
[THOUGHT] The Python output confirms n=25. [/THOUGHT] \boxed{25}
Explanation
The reasoning begins by analyzing the integral ∫ from a_n to a_{n+1} of f(x) dx, where a_n = 2^{n-1} - 1 and f(x) is piecewise-defined. On the interval [a_n, a_{n+1}) = [2^{n-1} - 1, 2^n - 1), f(x) = √[2^{n-1}(x + 1) - 2^{2n-2}] + 2^{n-1} - 1. Substituting k = 2^{n-1} simplifies the integral to (5/3)k^2 - k.
Setting this equal to the given value 1407374833221632 / 3 yields the equation 5k^2 - 3k = 1407374833221632, or 5k^2 - 3k - N = 0 with N = 1407374833221632. Solving the quadratic equation gives k = [3 + √(9 + 20N)] / 10.
The Python code computes the discriminant D = 9 + 20N = 28147496664432649, verifies it is a perfect square (√D ≈ 167772157, and 167772157^2 = D), then calculates p_num = 3 + 167772157 = 167772160. Dividing by 10 yields p = 16777216, which is confirmed as a power of 2 (2^24, since 16777216 & 16777215 == 0).
Thus, k = p = 2^24, so n-1 = 24, giving n = 25. The output directly prints 25, confirming the solution.