Question

Um atleta corre 400 metros a mais que no dia anterior. Ao final de 11 dias ele percorre um total de 35200 metros. Qual é o numero de metros que ele correu no ultimo dia?

Answer 1

a1 = x
a2 = x + 400 
a3 = x + 400 + 400 = x + 800
r = 400
a11 = x + 10r

S11 = 35.200
( x + x + 10r ).5,5 = 35200
( 2x + 10r).5,5 = 35200
( 2x + 10.400) = 35200/5,5
2x + 4000 =6400
2x = 6400 - 4000
2x = 2400
x = 1200 **  ( primeiro dia )

a11 =  x + 10r = 1200 + 10(400 ) = 1200 = 4000 = 5.200 m **** no último dia

Answer 2

$\sf \displaystyle 35200=\left(a1+an\right)\frac{11}{2}\\\\\\\sf 35200=\left(a\cdot \:1+an\right)\frac{11}{2}\\\\\\\sf \left(a\cdot \:1+an\right)\frac{11}{2}=35200\\\\\\\sf 2\left(a\cdot \:1+an\right)\frac{11}{2}=35200\cdot \:2\\\\\\\sf 11\left(a+an\right)=70400\\\\\\\sf \frac{11\left(a+an\right)}{11}=\frac{70400}{11}\\\\\\\sf a+an=6400\\\\\\\sf a\left(1+n\right)=6400\\\\\\\sf \frac{a\left(1+n\right)}{1+n}=\frac{6400}{1+n}\\\\\\\to \boxed{\boxed{\sf a=6400}}$

$\tt an=\left(6400-an\right)+\left(11-1\right)\cdot 400\\\\\\\tt an=6400-an+\left(11-1\right)\cdot \:400\\\\\\\tt an=6400-an+4000\\\\\\\tt an=-an+10400\\\\\\\tt an+an=-an+10400+an\\\\\\\tt 2an=10400\\\\\\\tt \dfrac{2an}{2n}=\dfrac{10400}{2n}\\\\\\\to \boxed{\boxed{\sf a=5200}}$