❗1. 8. 2026 (L) Fórum bude brzy uzavřeno 😿
Nejste přihlášen(a). Přihlásit
Stránky: 1
Offline
...
↑ stuart clark:
Thanks ↑ Pavel:
be the sum that is supposed to be found. Then![$
a_{n+1}-a_n
&=\sum_{s=0}^r(-1)^s\cdot(r-s+1)\cdot\left({n+1\choose s}-{n\choose s}\right)
=\sum_{s=1}^r(-1)^s\cdot(r-s+1)\cdot{n\choose s-1}\\[.5\baselineskip]
&=\sum_{s=0}^{r-1}(-1)^{s+1}\cdot(r-s)\cdot{n\choose s}
=\sum_{s=0}^r(-1)^{s+1}\cdot(r-s)\cdot{n\choose s}\\[.5\baselineskip]
&=\sum_{s=0}^r(-1)^{s+1}\cdot(r-s+1)\cdot{n\choose s}-\sum_{s=0}^r(-1)^{s+1}\cdot{n\choose s}\\[.5\baselineskip]
&=-\sum_{s=0}^r(-1)^s\cdot(r-s+1)\cdot{n\choose s}+\sum_{s=0}^r(-1)^s\cdot{n\choose s}\\[.5\baselineskip]
&=-a_n+\sum_{s=0}^r(-1)^s\cdot{n\choose s}.
$](/mathtex/4d/4d320124aa381fbd2d68d6047a5ec760.gif "kopírovat do textarea")
, thus![$
a_{n+2}-a_{n+1}
&=\sum_{s=0}^r(-1)^s\cdot\left({n+1\choose s}-{n\choose s}\right)
=\sum_{s=1}^r(-1)^s\cdot{n\choose s-1}
=\sum_{s=0}^{r-1}(-1)^{s+1}\cdot{n\choose s}\\[.5\baselineskip]
&=\sum_{s=0}^r(-1)^{s+1}\cdot{n\choose s}-(-1)^{r+1}{n\choose r}
=-\sum_{s=0}^r(-1)^s\cdot{n\choose s}+(-1)^r{n\choose r}\\[.5\baselineskip]
&=-a_{n+1}+(-1)^r{n\choose r}.
$](/mathtex/a6/a62a67501f5ea436e079b67c63b07e09.gif "kopírovat do textarea")
