Seja f (n) a soma dos n termos de uma progressão aritmética. Demonstrar que
f (n + 3) − 3 f (n + 2)+ 3 f (n + 1) − f (n) = 0 .
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Boa noite!
Demonstração
![f(n+1)=s+r\\\\\
f(n+2)=s+2r\\\\\
f(n+3)=s+3r](https://tex.z-dn.net/?f=f%28n%2B1%29%3Ds%2Br%5C%5C%5C%5C%5C%0Af%28n%2B2%29%3Ds%2B2r%5C%5C%5C%5C%5C%0Af%28n%2B3%29%3Ds%2B3r "f(n+1)=s+r\\\\\
f(n+2)=s+2r\\\\\
f(n+3)=s+3r")
![S+3r-3(S+2r)+3(S+r)-s=0\\\\\
S+3r-3S-6r+3S+3r-S=0\\\\\\
S-S-3S+3S+3r+3r-6r=0\\\\\\
-4S+4S+6r-6r=0\\\\\\
0+0=0\\\\\\
0=0\\\\\\
CQD](https://tex.z-dn.net/?f=S%2B3r-3%28S%2B2r%29%2B3%28S%2Br%29-s%3D0%5C%5C%5C%5C%5C%0AS%2B3r-3S-6r%2B3S%2B3r-S%3D0%5C%5C%5C%5C%5C%5C%0AS-S-3S%2B3S%2B3r%2B3r-6r%3D0%5C%5C%5C%5C%5C%5C%0A-4S%2B4S%2B6r-6r%3D0%5C%5C%5C%5C%5C%5C%0A0%2B0%3D0%5C%5C%5C%5C%5C%5C%0A0%3D0%5C%5C%5C%5C%5C%5C%0A%0ACQD%0A%0A "S+3r-3(S+2r)+3(S+r)-s=0\\\\\
S+3r-3S-6r+3S+3r-S=0\\\\\\
S-S-3S+3S+3r+3r-6r=0\\\\\\
-4S+4S+6r-6r=0\\\\\\
0+0=0\\\\\\
0=0\\\\\\
CQD")
Boa noite!
Bons estudos!
Demonstração
Boa noite!
Bons estudos!
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