If R = (6 √6 + 14)2n + 1 and f = R – \[R], where \[.] denote... | MATH - PROPERTIES OF BINOMIAL COEFFICIENTS | SolveeIt | Solveeit

Today, August 24

Question

If R = (6 √6 + 14)^{2n + 1} and f = R – [R], where [.] denotes the greatest integer function then R. f =

A

(20)ⁿ

B

(20)²ⁿ

C

(20)^{2n + 1}

D

None

Answer

(20)^{2n + 1}

Explanation

Solution

R = (6 $\sqrt{6}$ +14)²ⁿ⁼¹ =[R]+f=²ⁿ⁺¹C₀(6 $\sqrt{6}$ )²ⁿ⁺¹ +......

– (6 $\sqrt{6}$ – 14)²ⁿ⁺¹ = f = ²ⁿ⁺¹C₀(6 $\sqrt{6}$ )²ⁿ⁺¹ –.........

[R] + f – f ' = even Integer

Ž f – f ' = Integer

Ž f – f ' = 0

f = f '

Ž R. f = R f '

= (6 $\sqrt{6}$ +14)²ⁿ⁺¹. (6 $\sqrt{6}$ –14)²ⁿ⁺¹

= (216 – 169)²ⁿ⁺¹

= (20)²ⁿ⁺¹