The answer is 77 becuase you have to add all of the sides that are same as the number
Step-by-step explanation:

Given expression is

To, evaluate this limit, let we simplify numerator and denominator individually.
So, Consider Numerator

Clearly, if forms a Geometric progression with first term n and common ratio n respectively.
So, using Sum of n terms of GP, we get


Now, Consider Denominator, we have

can be rewritten as

![\rm \: = \: {n}^{n}\bigg[1 +\bigg[{\dfrac{n - 1}{n}\bigg]}^{n} + \bigg[{\dfrac{n - 2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B%7B%5Cdfrac%7Bn%20-%201%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7Bn%20-%202%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D)
![\rm \: = \: {n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D)
Now, Consider

So, on substituting the values evaluated above, we get
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\dfrac{ {n}^{n} - 1}{1 - \dfrac{1}{n} }}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%5Cdfrac%7B%20%7Bn%7D%5E%7Bn%7D%20%20-%201%7D%7B1%20-%20%20%5Cdfrac%7B1%7D%7Bn%7D%20%7D%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n} - 1}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%20%7Bn%7D%5E%7Bn%7D%20%20-%201%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n}\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7Bn%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7Bn%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D%7D%7B%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{1}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B1%7D%7B%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
Now, we know that,
![\red{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to \infty} \bigg[1 + \dfrac{k}{x} \bigg]^{x} = {e}^{k}}}}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%20%3A%5Clongmapsto%5C%3A%5Cboxed%7B%5Ctt%7B%20%5Cdisplaystyle%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cbigg%5B1%20%2B%20%5Cdfrac%7Bk%7D%7Bx%7D%20%5Cbigg%5D%5E%7Bx%7D%20%20%3D%20%20%7Be%7D%5E%7Bk%7D%7D%7D%7D%20)
So, using this, we get

Now, in denominator, its an infinite GP series with common ratio 1/e ( < 1 ) and first term 1, so using sum to infinite GP series, we have





Hence,

The Volume of air needed for a spherical balloon of radius 4 inch and circumference 8π is 85.3 cubic inch
<h3>Volume of Sphere</h3>
Given Data
Let us find radius
circumference = 2πr
8π = 2πr
8 = 2r
r = 8/2
r = 4 inch
Let us find the volume of the sphere
Volume = 4/3πr^3
Volume = 4/3* π*4^3
Volume = 4/3*π64
Volume = 256π/3
Volume = 85.3 cubic inch
Learn more about volume of sphere here:
brainly.com/question/22807400
10 - 5x < 6x - 8
We can treat this inequality like a normal algebraic equation.
<em><u>Add 5x to both sides</u></em>
10 < 11x - 8
<em><u>Add 8 to both sides</u></em>
18 < 11x
<em><u>Divide both sides by 11</u></em>
1.64 < x
X is greater than the value 1.64.
Let's plug the number 2 into each value of x to verify.
10 - 5x < 6x - 8
10 - 5*2) < (6 * 2) - 8
10 - 10 < 12 - 8
0 < 4
It's correct.
Now let's plug 1, which is less than 1.64.
10 - 5x < 6x - 8
10 - 5 < 6 - 8
5 < -2
This is not correct, which means that our answer was correct.
Answer:
BC = 24
Step-by-step explanation:
In the picture attached, the missing triangles are shown:
<u>Data</u>
As a consequence of the "Side Splitter" Theorem:
AD/DE = AB/BC
Replacing with data and solving for BC
5/12 = 10/BC
BC = 10*12/5 = 24