First slide

Binomial theorem for positive integral Index

Question

lntegral part of $(7 + 4 \sqrt{3})^{n} if (n \in N)$

Moderate

A

an even number
B

an odd number
C

an even or an odd number depending upon the value of n
D

None of the above

Solution

$Here, \forall n \in N, (7 + 4 \sqrt{3})^{n} \notin N$
$∴ Denote (7 + 4 \sqrt{3})^{n} by l + f$
$where, l is an integer and f \in R such that 0 < f < 1$
$∵ 0 < 7 - 4 \sqrt{3} < 1$
$∴ We can denote (7 - 4 \sqrt{3})^{n} by C where, C \in R suchthat 0 < C < 1,$
$Now, I + f = (7 + 4 \sqrt{3})^{n} = 7^{n} +^{n} C_{1} 7^{n - 1} (4 \sqrt{3})$ $+^{n} C_{2} 7^{n - 2} (4 \sqrt{3})^{2} + \dots$ (i)
$C = (7 - 4 \sqrt{3})^{n} = 7^{n} -^{n} C_{1} 7^{n - 1} (4 \sqrt{3}) +^{n} C_{2} 7^{n - 2} (4 \sqrt{3})^{2} \dots$ (ii)
$To cancel irrational terms we add Eqs . (i) and (ii), weget$
$1 + f + C = 2 (7^{n} +^{n} C_{2} 7^{n - 2} (48) +^{n} C_{4} 7^{n - 4} (48)^{2} + \dots)$
$= 2 k, where k is an integer$ …(iii)
Now, 0<f<1
and 0 <C<1
$\Rightarrow 0 < f + C < 2$ ...(iv)
Form Eqs. (iii) and 1iv), f + C -1
Now, form Eq. (iii) l=2k-1
i.e., I is an odd integer.

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