2. [latex]b_n= frac{4-n}{10}[/latex] Liczymy [latex]b_{n+1}[/latex] [latex]b_{n+1}= frac{4-(n+1)}{10}[/latex] [latex]b_{n+1}= frac{4-n-1}{10}[/latex] [latex]b_{n+1}= frac{3-n}{10}[/latex] Liczymy [latex]b_{n+1}-b_n[/latex] [latex]b_{n+1}-b_n= frac{3-n}{10}-frac{4-n}{10}= frac{3-n-(4-n)}{10}=[/latex] [latex]frac{3-n-4+n}{10}=frac{-1}{10}<0[/latex] - ciąg jest malejący 3. [latex]a_1=5[/latex] [latex]r=4[/latex] [latex]a_{15}=a_1+14r[/latex] [latex]a_{15}=5+14cdot4=5+56=61[/latex] [latex]S_{n}=[a_1+ frac{(n-1)r}{2}]n[/latex] [latex]S_{20}=[5+ frac{(20-1)cdot 4}{2}]cdot 20[/latex] [latex]S_{20}=[5+frac{19cdot 4}{2}]cdot 20[/latex] [latex]S_{20}=[5+frac{76}{2}]cdot 20[/latex] [latex]S_{20}=[5+38]cdot 20[/latex] [latex]S_{20}=43cdot 20[/latex] [latex]S_{20}=860[/latex] 5. [latex]a_{17}=-7[/latex] [latex]a_{26}=-25[/latex] Wyznaczam [latex]a_1[/latex] [latex]a_{17}=a_1+16r[/latex] [latex]a_1+16r=-7[/latex] [latex]a_1=-7-16r[/latex] Obliczam r [latex]a_{26}=a_1+25r[/latex] [latex]a_{26}=-7-16r+25r[/latex] [latex]a_{26}=9r-7[/latex] [latex]9r-7=-25[/latex] [latex]9r=-25+7[/latex] [latex]9r=-18[/latex] [latex]r=-2[/latex] Obliczam a_1 [latex]a_1=-7-16r[/latex] [latex]a_1=-7-16cdot (-2)[/latex] [latex]a_1=-7+32[/latex] [latex]a_1=25[/latex]
