Binomial theorem for positive integral Index

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Question

Least positive integer just greater than $(1 + 0 .00002)^{50000}$ is __________.

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Solution

$(1 + 0 .00002)^{50000} = {(1 + \frac{1}{50000})}^{50000}$ $= 1 + 50000 \times \frac{1}{50000} + 50000_{c_{2}} \frac{1}{{(50000)}^{2}}$ +…….
$\frac{1}{{(50000)}^{2}} i s v e r y s m a l l s o n e g l e c t h i g h e r p o w e r s$
Now we know that $2 \leq {(1 + \frac{1}{n})}^{n} < 3 \forall n \geq 1$
$\Rightarrow Least integer is 3$

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

Least positive integer just greater than (1+0.00002)50000 is __________.

(1+0.00002)50000=1+15000050000=1+50000×150000+50000c21(50000)2+…….1(50000)2is very small so neglect higher powersNow we know that 2≤1+1nn<3∀n≥1⇒ Least integer is 3

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Least positive integer just greater than $(1 + 0 .00002)^{50000}$ is __________.