if there is just supposed to be a one-word answer, I think it's
<h2>true</h2>
Answer:
I believe its 30
Step-by-step explanation:
because he studied for five minutes the first day and then added five minutes per day for 6 days so i think its 30
Via the generating function method, let
![G(x)=\displaystyle\sum_{n\ge0}a_nx^n](https://tex.z-dn.net/?f=G%28x%29%3D%5Cdisplaystyle%5Csum_%7Bn%5Cge0%7Da_nx%5En)
Then take the recurrence,
![a_n=6a_{n-1}-8a_{n-2}](https://tex.z-dn.net/?f=a_n%3D6a_%7Bn-1%7D-8a_%7Bn-2%7D)
multiply everything by
![x^n](https://tex.z-dn.net/?f=x%5En)
and sum over all
![n\ge2](https://tex.z-dn.net/?f=n%5Cge2)
:
![\displaystyle\sum_{n\ge2}a_nx^n=6\sum_{n\ge2}a_{n-1}x^n-8\sum_{n\ge2}a_{n-2}x^n](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%5Cge2%7Da_nx%5En%3D6%5Csum_%7Bn%5Cge2%7Da_%7Bn-1%7Dx%5En-8%5Csum_%7Bn%5Cge2%7Da_%7Bn-2%7Dx%5En)
Re-index the sums or add/remove terms as needed in order to be able to express them in terms of
![G(x)](https://tex.z-dn.net/?f=G%28x%29)
:
![\displaystyle\sum_{n\ge2}a_nx^n=\sum_{n\ge0}a_nx^n-(a_0-a_1x)=G(x)-4-10x](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%5Cge2%7Da_nx%5En%3D%5Csum_%7Bn%5Cge0%7Da_nx%5En-%28a_0-a_1x%29%3DG%28x%29-4-10x)
![\displaystyle\sum_{n\ge2}a_{n-1}x^n=\sum_{n\ge1}a_nx^{n+1}=x\sum_{n\ge1}a_nx^n=x\left(G(x)-a_0\right)=x(G(x)-4)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%5Cge2%7Da_%7Bn-1%7Dx%5En%3D%5Csum_%7Bn%5Cge1%7Da_nx%5E%7Bn%2B1%7D%3Dx%5Csum_%7Bn%5Cge1%7Da_nx%5En%3Dx%5Cleft%28G%28x%29-a_0%5Cright%29%3Dx%28G%28x%29-4%29)
![\displaystyle\sum_{n\ge2}a_{n-2}x^n=\sum_{n\ge0}a_nx^{n+2}=x^2\sum_{n\ge0}a_nx^n=x^2G(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%5Cge2%7Da_%7Bn-2%7Dx%5En%3D%5Csum_%7Bn%5Cge0%7Da_nx%5E%7Bn%2B2%7D%3Dx%5E2%5Csum_%7Bn%5Cge0%7Da_nx%5En%3Dx%5E2G%28x%29)
So the recurrence relation is transformed to
![G(x)-4-10x=6x(G(x)-4)-8x^2G(x)](https://tex.z-dn.net/?f=G%28x%29-4-10x%3D6x%28G%28x%29-4%29-8x%5E2G%28x%29)
![(1-6x+8x^2)G(x)=4-14x](https://tex.z-dn.net/?f=%281-6x%2B8x%5E2%29G%28x%29%3D4-14x)
![G(x)=\dfrac{4-14x}{1-6x+8x^2}=\dfrac{4-14x}{(1-4x)(1-2x)}=\dfrac1{1-4x}+\dfrac3{1-2x}](https://tex.z-dn.net/?f=G%28x%29%3D%5Cdfrac%7B4-14x%7D%7B1-6x%2B8x%5E2%7D%3D%5Cdfrac%7B4-14x%7D%7B%281-4x%29%281-2x%29%7D%3D%5Cdfrac1%7B1-4x%7D%2B%5Cdfrac3%7B1-2x%7D)
For appropriate values of
![x](https://tex.z-dn.net/?f=x)
, we can express the RHS in terms of geometric power series:
![G(x)=\displaystyle\sum_{n\ge0}(4x)^n+3\sum_{n\ge0}(2x)^n=\sum_{n\ge0}\bigg(4^n+3\cdot2^n\bigg)x^n](https://tex.z-dn.net/?f=G%28x%29%3D%5Cdisplaystyle%5Csum_%7Bn%5Cge0%7D%284x%29%5En%2B3%5Csum_%7Bn%5Cge0%7D%282x%29%5En%3D%5Csum_%7Bn%5Cge0%7D%5Cbigg%284%5En%2B3%5Ccdot2%5En%5Cbigg%29x%5En)
which tells us that