First slide

Binomial theorem for positive integral Index

Question

If $a = 99^{50} + 100^{50}$ and $b = 101^{50}$ then

Moderate

A

a < b
B

a = b
C

a > b
D

a – b = 100⁴⁹

Solution

$\begin{array}{l} 101^{50} - 99^{50} = (100 + 1)^{50} - (100 - 1)^{50} \\ = 2 [^{50} C_{1} (100)^{49} +^{50} C_{3} (100)^{47} + \dots +^{50} C_{49} (100)] \end{array}$ \
$= 100^{50} +$ a positive term.
$\Rightarrow b = a +$ (a positive term) $\Rightarrow a < b .$

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