Répondre :
S(n)=9+99+999+...+9999...999 (n fois)
=9 [1+(1+10)+(1+10+100)+...+(1+10+100+...10^(n-1)]
=9[(1-10)/(1-10)+(1-10²)/(1-10)+(1-10³)/(1-10)+...+(1-10^n)/(1-10)]
=(-1)[(1-10)+(1-10²)+(1-10³)+...+(1-10^n)]
=(-1)[(1+1+1+...+1)-(10+10²+10³+10^n)]
=-n+10(1+10+10²+...+10^(n-1)]
=-n+10(1-10^n)/(1-10)
=-n-10/9(1-10^n)
=1/9*10^(n+1)-n-10/9
=9 [1+(1+10)+(1+10+100)+...+(1+10+100+...10^(n-1)]
=9[(1-10)/(1-10)+(1-10²)/(1-10)+(1-10³)/(1-10)+...+(1-10^n)/(1-10)]
=(-1)[(1-10)+(1-10²)+(1-10³)+...+(1-10^n)]
=(-1)[(1+1+1+...+1)-(10+10²+10³+10^n)]
=-n+10(1+10+10²+...+10^(n-1)]
=-n+10(1-10^n)/(1-10)
=-n-10/9(1-10^n)
=1/9*10^(n+1)-n-10/9