(5+1)^4+(5-1)^4 is - Vidyadip

MCQ

$(\sqrt{5}+1)^4+(\sqrt{5}-1)^4$ is

A
an irrational number
B
a negative real number
✓
a rational number
D
a negative integer

✓

Answer

Correct option: C.

a rational number

We have $( a + b )^{ n }+( a - b )^{ n }$
$=\left[{ }^n C_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2+{ }^n C_3 a^{n-3} b^3+\ldots . .+{ }^n C_n b^n\right]+$
${\left[{ }^n C_0 a^n-{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2-{ }^n C_3 a^{n-3} b^3+\ldots . .+(-1)^n \cdot{ }^n C_n b^n\right]}$
$=2\left[{ }^n C_0 a^n+{ }^n C_2 a^{n-2} b^2+\ldots\right]$
Let $a=\sqrt{5}$ and $b=1$ and $n=4$
Now we get $(\sqrt{5}+1)^4+(\sqrt{5}-1)^4=2\left[{ }^4 C_0(\sqrt{5})^4+{ }^4 C_2(\sqrt{5})^2 1^2+{ }^4 C_4(\sqrt{5})^0 1^4\right]$
$=2[25+30+1]=112$

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