Question

Integral part of (5$\sqrt{5}$+ 11)^{2n + 1} is –

• A. Even
• B. Odd
• C. Even or odd depending upon the value of n
• D. Cannot be determined

Answer 1

Answer: (A)

Even

Answer 2

(5$\sqrt{5}$ + 11)^{2n + 1} = I + f

Now suppose F = (5$\sqrt{5}$ – 11)^{2n + 1}

I + f – F = (5$\sqrt{5}$ + 11)^{2n + 1} – (5$\sqrt{5}$ – 11)^{2n + 1}

= 2 [^{2n + 1}C₁ (125)ⁿ + ^{2n + 1}C₃ (125)^{n – 1} 11³ + …]

= 2 even integer

I + f – F = 2k

f – F = 2k – I is an integer

0 £ f < 1 0 < F < 1– 1 < – F < 0– 1 < f – F < 1

Ž f – F = 0

I = 2k Ž even integer