If n is a positive integer, then ({3}+1)²n}-({3}-1)²n} is : [AIEEE 2012]

Correct Answer is : an irrational number

a rational number other than positive integers

Correct Answer is : an irrational number

Test Index

Binomial Theorem Part 1

Section: Mathematics

Question : 78 of 100

Marks: +1, -0

n

(\sqrt{3}+1)^{2n}-(\sqrt{3}-1)^{2n}

Solution:

Let

(a+x)^n=

Odd trems(A) + Even terms

(B)

(a-x)^n=

Odd terms(A) - Even terms(B)

\; \therefore (a+x)^n-(a-x)^n

\;=(A+B)-(A-B)

\;=2B

\;=2[\;\text{ even terms] }\;

\;=2 [ T_2+ T_4+ T_6+ \cdots ]

So in case of

(\sqrt{3}+1)^{2n}-(\sqrt{3}-1)^{2n}

=2 [ T_2+T_4+T_6+ \cdots ]

=2 \left[ {}^{2n}C_1 \cdot (\sqrt{3})^{2n-1} + {}^{2n}C_3 \cdot (\sqrt{3})^{2n-3} + \cdots \right]

Here in

(\sqrt{3})^{2n-1}, 2n-1

is odd number. So there will be always

\sqrt{3}

(\sqrt{3})^{2n-1}

. So

(\sqrt{3}+1)^{2n}-(\sqrt{3}-1)^{2n}

will be always irrational number.

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