Question

$(3\sqrt{3}+5{)}^n$ = p + f, where p is an integer and f is a proper fraction(fractional part of $(3\sqrt{3}+5{)}^n$). Find the value of $(3\sqrt{3}-5{)}^n$ in terms of p and f, if n is an even integer.

• A. p
• B. 1 - f
• C. p - f
• D. f

Answer 1

The correct option is B

1 - f

It is given that $(3\sqrt{3}+5{)}^n$ = p + f. Consider $(3\sqrt{3}-5{)}^n$. This value lies between 0 and 1, because

$(3\sqrt{3}-5)$ lies between 0 and 1.

Let $(3\sqrt{3}-5{)}^n$ = f'

We will expand both $(3\sqrt{3}+5{)}^n$ and $(3\sqrt{3}-5{)}^n$.

p + f = $(3\sqrt{3}+5{)}^n$ = $(3\sqrt{3}{)}^n$ + ${}^n{C}_1$ $(3\sqrt{3}{)}^{n-1}$$(5{)}^1$ + ${}^n{C}_2$ $(3\sqrt{3}{)}^{n-2}$$(5{)}^2$ ......................(1)

f' = $(3\sqrt{3}-5{)}^n$ = $(3\sqrt{3}{)}^n$ - ${}^n{C}_1$ $(3\sqrt{3}{)}^{n-1}$$(5{)}^1$ + ${}^n{C}_2$ $(3\sqrt{3}{)}^{n-2}$$(5{)}^2$ ...................(2)

We have to decide if we should add (1) and (2) or subtract. We will perform the operation which will

remove the radical sign. $(3\sqrt{3}{)}^n$ will be radical free because, n is an even integer. So we want to retain $(3\sqrt{3}{)}^n$

after the operation ⇒ we will perform addition.

⇒ p + f + f' = $(3\sqrt{3}+5{)}^n$ + $(3\sqrt{3}-5{)}^n$ = 2[$(3\sqrt{3}{)}^n$ + ${}^n{C}_2$$(3\sqrt{3}{)}^{n-2}$$(5{)}^2$.............] ............................(3)

⇒ p + f + f' is an even integer (from(3)).

p is also an integer. For p + f + f' to be an integer, f + f' should also be an integer. The only integer value it

can take is 1, because 0 < f + f' < 2.

⇒ f + f' = 1

⇒ f' = 1 - f