Question

Calculati sumele :
a) 1 + 4 + 7 + 10 + ... + 541 ;
b) 1 + 5 + 9 + .... + 725 ;
c) 3 + 7 + 11 + 15 + .... + 743
d) 5 + 7 + 9 + 11 + ... + 2005

Answer 1

    
[tex]Notatie: \\ \text{n = numarul de termeni pe care il vom calcula la } \\ \text{fiecare din punctele a) b) c) d).} \\ \\ \displaystyle a) \\ n= \frac{541-1}{3}+1 = \frac{540}{3}+1 =180+1 = 181 \\ \\ S = 1 + 4 + 7 + 10 + ... + 541 = \\ \\ =\frac{181(541+1)}{3-1}= \frac{181(542)}{2}=181\times 271 = \boxed{49051} [/tex]


[tex]\displaystyle b) \\ n= \frac{725-1}{4}+1 = \frac{724}{4}+1 + 181+1=182 \\ \\ S=1 + 5 + 9 + .... + 725= \frac{182(725+1)}{2} =91\times 726 =\boxed{66066} \\ \\ \\ c) \\ n= \frac{743-3}{4}+1 =\frac{740}{4}+1=185+1 = 186 \\ \\ S=3 + 7 + 11 + 15 + .... + 743= \frac{186(743+3)}{2}=93 \times 746 = \boxed{69378}[/tex]


[tex]\displaystyle d) \\ n= \frac{2005-5}{2}+1 = \frac{2000}{2}+1 =1000+1=1001 \\ \\ S=5 + 7 + 9 + 11 + ... + 2005 = \\ \\ =\frac{1001(2005+5)}{2}= \frac{1001\times 2010}{2}=1001\times 1005 = \boxed{1006005}[/tex]