Could they be the same?

Calculus Level 1

Which is bigger?

(1.1)^{10000}

1000

2 solutions

Arjen Vreugdenhil
Oct 11, 2015

The trick is to observe that $(1.1)^n = 1 + n\cdot(0.1) + \dots > 1 + \frac{n}{10}.$ Therefore, $(1.1)^{10000} > 1 + 1000.$

Manish Kumar Singh
Oct 12, 2015

we know that (1+x)^n= 1+[nx]+[n(n-1)/2] *x^2 --------------; now ,(1.1)= (1+0.1); (1.1)^10000 =(1+0.1)^10000; =1+ (10000 X 0.1)+ a positive value; =1+1000+ a positive value; =1001+ a positive value > 1000

Your solution in Latex:

${ \left( 1+x \right) }^{ n }=1+nx+\frac { { n(n-1)x }^{ 2 } }{ 2! } +\frac { { n(n-1)(n-2)x }^{ 3 } }{ 3! } +\dots \\ \\ \quad \quad \quad \quad \quad { (1.1) }^{ 10000 }={ (1+0.1) }^{ 10000 }\\ \quad \quad \quad \quad \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =1+10000\times 0.1+\dots \\ \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad =1+1000+\dots \\ \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =1001+\dots \quad >\quad 1000$

Dinesh Nath Goswami - 5 years, 8 months ago

thanks ,next time i will write in Latex.

manish kumar singh - 5 years, 8 months ago

Could they be the same?

2 solutions

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