Question

Which of the following values is closest to k when k=(1+\frac{1}{n})^n, given n is equal to the number of seconds in one year? e 2.7 2 2.75

Answer 1

The value closest to k is e = 2.72.

Step-by-step explanation:

We are given the following expression  $k = (1+\frac{1}{n})^{n}$ , where n is equal to the number of seconds in one year.

And we have to find the value that is closest to k.

Firstly, as it is given that n is equal to the number of seconds in one year;

So, the number of seconds(n) in one year is = $365 \times 24 \times 60 \times 60$

                                                                        =  3,15,36,000 seconds.

where; 365 = number of days in one year (ignoring leap year)

              24 = number of hours in one day

              60 = number of minutes in one hour  

              60 = number of seconds in one minute

Now, the given expression is;

$k = (1+\frac{1}{n})^{n}$

Substituting the value of n in the above expression we get;

$k = (1+\frac{1}{31536000})^{31536000}$

Now, as we know that  $\lim_{x \to \infty} (1+\frac{1}{x})^{x} = e$ .

Similarly, here we can see that 32536000 is also approaching to infinity as it is also a very large number, that means;

k is approximately equal to e, i.e; k ≈ e and the value of e is 2.72.

Hence, the value closest to k is e = 2.72.

Answer 2

k is closest to e  = 2.72

Step-by-step explanation:

Given

$k=(1+\frac{1}{n})^n$

Where n is the number of seconds in one year

Number of seconds in a year

$=365\times 24\times 60\times 60$

$=32061600$

Thus,

$k=(1+\frac{1}{32061600})^{32061600}$

We know that

$\lim_{n \to \infty} (n+\frac{1}{n})^n=e$

Since 32061600 is sufficiently large number

Therefore, we can say that

$k\approx e$

or $k\approx 2.72$

Hope this answer is helpful.

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