If the nth term is 3n+4, then the simplified (n+1)st term would be 3n+5. (True or False)
1 answer:
Answer:
![\large \boxed{\sf False}](https://tex.z-dn.net/?f=%5Clarge%20%5Cboxed%7B%5Csf%20False%7D)
Step-by-step explanation:
![3n+4](https://tex.z-dn.net/?f=3n%2B4)
Replace the n with n+1 to find the (n+1)st term.
![3(n+1)+4](https://tex.z-dn.net/?f=3%28n%2B1%29%2B4)
Expand brackets.
![3n+3+4](https://tex.z-dn.net/?f=3n%2B3%2B4)
Add like terms.
![3n+7](https://tex.z-dn.net/?f=3n%2B7)
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![r=\theta\sin\theta+\cos\theta](https://tex.z-dn.net/?f=r%3D%5Ctheta%5Csin%5Ctheta%2B%5Ccos%5Ctheta)
![\dfrac{\mathrm dr}{\mathrm d\theta}=\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}+\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D%5Cdfrac%7B%5Cmathrm%20d%28%5Ctheta%5Csin%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D%2B%5Cdfrac%7B%5Cmathrm%20d%28%5Ccos%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D)
By the product rule,
![\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}=\dfrac{\mathrm d(\theta)}{\mathrm d\theta}\sin\theta+\theta\dfrac{\mathrm d(\sin\theta)}{\mathrm d\theta}=\sin\theta+\theta\cos\theta](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%28%5Ctheta%5Csin%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D%5Cdfrac%7B%5Cmathrm%20d%28%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D%5Csin%5Ctheta%2B%5Ctheta%5Cdfrac%7B%5Cmathrm%20d%28%5Csin%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D%5Csin%5Ctheta%2B%5Ctheta%5Ccos%5Ctheta)
and
![\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}=-\sin\theta](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%28%5Ccos%5Ctheta%29%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D-%5Csin%5Ctheta)
So we have
![\dfrac{\mathrm dr}{\mathrm d\theta}=\sin\theta+\theta\cos\theta-\sin\theta](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D%5Csin%5Ctheta%2B%5Ctheta%5Ccos%5Ctheta-%5Csin%5Ctheta)
![\implies\dfrac{\mathrm dr}{\mathrm d\theta}=\theta\cos\theta](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20d%5Ctheta%7D%3D%5Ctheta%5Ccos%5Ctheta)
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