Answer:
![\displaystyle \large{\sum_{n=1}^{42}(6n-1)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B%5Csum_%7Bn%3D1%7D%5E%7B42%7D%286n-1%29)
Step-by-step explanation:
Given:
To find:
- Summation notation of the given series
Summation Notation:
![\displaystyle \large{\sum_{k=1}^n a_k}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B%5Csum_%7Bk%3D1%7D%5En%20a_k%7D)
Where n is the number of terms and
is general term.
First, determine what kind of series it is, there are two main series that everyone should know:
A series that has common difference.
A series that has common ratio.
If you notice and keep subtracting the next term with previous term:
Two common difference, we can in fact say that the series is arithmetic one. Since we know the type of series, we have to find the number of terms.
Now that brings us to arithmetic sequence, we know that first term is 5 and last term is 251, we’ll be finding both general term and number of term using arithmetic sequence:
<u>Arithmetic Sequence</u>
![\displaystyle \large{a_n=a_1+(n-1)d}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7Ba_n%3Da_1%2B%28n-1%29d%7D)
Where
is the nth term,
is the first term and
is the common difference:
So for our general term:
![\displaystyle \large{a_n=5+(n-1)6}\\\displaystyle \large{a_n=5+6n-6}\\\displaystyle \large{a_n=6n-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7Ba_n%3D5%2B%28n-1%296%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7Ba_n%3D5%2B6n-6%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7Ba_n%3D6n-1%7D)
And for number of terms, substitute
= 251 and solve for n:
![\displaystyle \large{251=6n-1}\\\displaystyle \large{252=6n}\\\displaystyle \large{n=42}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B251%3D6n-1%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7B252%3D6n%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7Bn%3D42%7D)
Now we can convert the series to summation notation as given the formula above, substitute as we get:
![\displaystyle \large{\sum_{n=1}^{42}(6n-1)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B%5Csum_%7Bn%3D1%7D%5E%7B42%7D%286n-1%29)
They are the same measurements, because:
from 10x + 5, first do vertical angle.
then, alternate interior angle,
then vertical angle again,
and another alternate interior angle.
For each vertical and alternate interior angles, the measurements stay the same.
Set them equal to each other, isolate and solve for x.
10x + 5 = 11x - 1
First, isolate the x. Subtract 10x from both sides, and add 1 to both sides
10x (-10x) + 5 (+1) = 11x (-10x) -1 (+1)
Simplify
5 + 1 = 11x - 10x
6 = x
x = 6
6 is your answer for x.
hope this helps
Answer:
The 38th term of 459,450,441,.. will be:
![a_{13}=351](https://tex.z-dn.net/?f=a_%7B13%7D%3D351)
Step-by-step explanation:
Given the sequence
![459,450,441,..](https://tex.z-dn.net/?f=459%2C450%2C441%2C..)
An arithmetic sequence has a constant difference 'd' and is defined by
![a_n=a_1+\left(n-1\right)d](https://tex.z-dn.net/?f=a_n%3Da_1%2B%5Cleft%28n-1%5Cright%29d)
computing the differences of all the adjacent terms
![450-459=-9,\:\quad \:441-450=-9](https://tex.z-dn.net/?f=450-459%3D-9%2C%5C%3A%5Cquad%20%5C%3A441-450%3D-9)
so
![d=-9](https://tex.z-dn.net/?f=d%3D-9)
The first element of the sequence is
![a_1=459](https://tex.z-dn.net/?f=a_1%3D459)
so the nth term will be
![a_n=-9\left(n-1\right)+459](https://tex.z-dn.net/?f=a_n%3D-9%5Cleft%28n-1%5Cright%29%2B459)
![a_n=-9n+468](https://tex.z-dn.net/?f=a_n%3D-9n%2B468)
Putting n=38 to find the 38th term
![a_n=-9n+468](https://tex.z-dn.net/?f=a_n%3D-9n%2B468)
![a_{13}=-9\left(13\right)+468](https://tex.z-dn.net/?f=a_%7B13%7D%3D-9%5Cleft%2813%5Cright%29%2B468)
![a_{13}=-117+468](https://tex.z-dn.net/?f=a_%7B13%7D%3D-117%2B468)
![a_{13}=351](https://tex.z-dn.net/?f=a_%7B13%7D%3D351)
Therefore, the 38th term of 459,450,441,.. will be:
![a_{13}=351](https://tex.z-dn.net/?f=a_%7B13%7D%3D351)