Question

4+9+14+19............+1004=

Answer 1

=(5*1-1)+(5*2-1)+...+(5*201-1)=5(1+2+...+201)-201=5*201*202/2-201=101304

Answer 2

$\displaystyle 4+9+14+19+...+1004 \\ \\ 1004=4+(n-1) \times 5 \\ \\ 1004=4+5n-5 \\ \\ 5n=1004-4+5 \\ \\ 5n=1005 \\ \\ n= \frac{1005}{5} \\ \\ n=201$

$\displaystyle S_{201}= \frac{8+200 \times 5}{2} \times 201 \\ \\ S_{201}= \frac{8+1000}{2} \times 201 \\ \\ S_{201}= \frac{1008}{2} \times 201 \\ \\ S_{201}=504 \times 201 \\ \\ S_{201}=101304$