JEEAdvanced
Find the greatest integer not exceeding (2 + √5)^5; that is, compute…
Question
Find the greatest integer not exceeding (2 + √5)^5; that is, compute ⌊(2 + √5)^5⌋.
✓ Verified answer: 1364checked by our engine — not a guess
Step-by-step solution
Let N = (2+√5)^5 and M = (2-√5)^5. Note 2-√5 ≈ -0.2360679..., so M is negative with |M| ≈ 0.000733 < 1; in fact M ∈ (-1, 0).
N + M is an integer because the irrational (√5) terms cancel by the binomial expansion: N + M = 2·[ C(5,0)2^5 + C(5,2)2^3·5 + C(5,4)2·25 ] = 2·(32 + 320 + 250) = 1364.
So N = 1364 - M, and since M ∈ (-1,0), we have N = 1364 + |M| with 0 < |M| < 1. Hence ⌊N⌋ = 1364.
Final answer1364
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Algebra solutions
Let a, b, c be the (possibly complex) roots of the cubic x^3 - 5x^2 +…JEE · MathA function f defined for all nonzero real x satisfies f(x) + 2 f(1/x)…JEE · MathLet a, b, c be positive real numbers with abc = 8. Find the minimum…JEE · MathA polynomial P(x) leaves remainder 3 on division by (x - 1),…JEE · MathThe cubic x^3 - a x^2 + (a - 1)x - 5 = 0 (with a a real parameter)…JEE · MathLet w be the real number w = cbrt(20 + 14√2) + cbrt(20 - 14√2), where…JEE · MathEvaluate the sum S = 1·1! + 2·2! + 3·3! + 4·4! + 5·5! + 6·6!, where…JEE · MathReal numbers a, b, c satisfy a + b + c = 6, a^2 + b^2 + c^2 = 14, and…JEE · Math