Answer:
![a_{103}=195](https://tex.z-dn.net/?f=a_%7B103%7D%3D195)
Step-by-step explanation:
<u>Sum of the first n terms of an arithmetic series formul</u>a:
![\large\boxed{S_n=\dfrac12n\left[2a+(n-1)d\right]}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7BS_n%3D%5Cdfrac12n%5Cleft%5B2a%2B%28n-1%29d%5Cright%5D%7D)
where:
- a is the first term.
- d is the common difference between terms.
Given:
![S_n=n^2-10n](https://tex.z-dn.net/?f=S_n%3Dn%5E2-10n)
Substitute the given equation into the formula and rearrange:
![\begin{aligned}\implies n^2-10n & =\dfrac12n[2a+(n-1)d]\\n(n-10)& = \dfrac{1}{2}n\left[2a+dn-d\right]\\n-10_&=\dfrac{1}{2}[2a+dn-d]\\2n-20& = 2a+dn-d\\ 2n-20 & = dn-(d-2a) \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cimplies%20n%5E2-10n%20%26%20%3D%5Cdfrac12n%5B2a%2B%28n-1%29d%5D%5C%5Cn%28n-10%29%26%20%3D%20%5Cdfrac%7B1%7D%7B2%7Dn%5Cleft%5B2a%2Bdn-d%5Cright%5D%5C%5Cn-10_%26%3D%5Cdfrac%7B1%7D%7B2%7D%5B2a%2Bdn-d%5D%5C%5C2n-20%26%20%3D%202a%2Bdn-d%5C%5C%202n-20%20%26%20%3D%20dn-%28d-2a%29%20%5Cend%7Baligned%7D)
Therefore:
![\implies 2n =dn](https://tex.z-dn.net/?f=%5Cimplies%202n%20%3Ddn)
![\implies d = 2](https://tex.z-dn.net/?f=%5Cimplies%20d%20%3D%202)
Therefore:
![\implies -20=-(d-2a)](https://tex.z-dn.net/?f=%5Cimplies%20-20%3D-%28d-2a%29)
![\implies 20=d-2a](https://tex.z-dn.net/?f=%5Cimplies%2020%3Dd-2a)
Substitute the found value of d:
![\implies 20=2-2a](https://tex.z-dn.net/?f=%5Cimplies%2020%3D2-2a)
![\implies 18=-2a](https://tex.z-dn.net/?f=%5Cimplies%2018%3D-2a)
![\implies 2a=-18](https://tex.z-dn.net/?f=%5Cimplies%202a%3D-18)
![\implies a=-9](https://tex.z-dn.net/?f=%5Cimplies%20a%3D-9)
Check the found values of a and d by substituting them into the sum formula and rearranging:
![\begin{aligned}\implies S_n & =\dfrac12n[2a+(n-1)d]\\& =\dfrac12n[2(-9)+(n-1)(2)]\\& =\dfrac12n[-18+2n-2]\\& =n[-9+n-1]\\& =-9n+n^2-n\\& = n^2-10n\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cimplies%20S_n%20%26%20%3D%5Cdfrac12n%5B2a%2B%28n-1%29d%5D%5C%5C%26%20%3D%5Cdfrac12n%5B2%28-9%29%2B%28n-1%29%282%29%5D%5C%5C%26%20%3D%5Cdfrac12n%5B-18%2B2n-2%5D%5C%5C%26%20%3Dn%5B-9%2Bn-1%5D%5C%5C%26%20%3D-9n%2Bn%5E2-n%5C%5C%26%20%3D%20n%5E2-10n%5Cend%7Baligned%7D)
As the equation matches the given equation, this verifies that:
- First term = -9
- Common difference = 2
<u>General form of an arithmetic sequence</u>:
![\large\boxed{a_n=a+(n-1)d}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7Ba_n%3Da%2B%28n-1%29d%7D)
where:
is the nth term.- a is the first term.
- d is the common difference between terms.
To find the 103rd term, substitute the found values of a and d together with n = 103 into the formula:
![\implies a_{103}=-9+(103-1)(2)](https://tex.z-dn.net/?f=%5Cimplies%20a_%7B103%7D%3D-9%2B%28103-1%29%282%29)
![\implies a_{103}=-9+(102)(2)](https://tex.z-dn.net/?f=%5Cimplies%20a_%7B103%7D%3D-9%2B%28102%29%282%29)
![\implies a_{103}=-9+204](https://tex.z-dn.net/?f=%5Cimplies%20a_%7B103%7D%3D-9%2B204)
![\implies a_{103}=195](https://tex.z-dn.net/?f=%5Cimplies%20a_%7B103%7D%3D195)
Therefore, the 103rd term of the arithmetic sequence is 195.
Learn more about arithmetic sequences here:
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