Question

Ache a soma dos n primeiros números da PA (2x + 1, 2x + 3,...)

Answer 1

Achando a razão da P.A:

$r=a_{2}-a_{1}\\r=(2x+3)-(2x+1)\\r=2x+3-2x-1\\r=2$

Achando o n-ésimo termo da P.A:

$a_{n}=a_{1}+(n-1)r\\a_{n}=(2x+1)+(n-1)\cdot2\\a_{n}=2x+1+2n-2\\a_{n}=2x+2n-1$

Achando a soma dos n primeiros termos da P.A:

$S_{n}=(a_{1}+a_{n})\cdot\dfrac{n}{2}\\\\\\S_{n}=(2x+1+2x+2n-1)\cdot\dfrac{n}{2}\\\\\\S_{n}=(4x+2n)\cdot\dfrac{n}{2}$

Colocando 2 em evidência:

$S_{n}=2\cdot(2x+n)\cdot\dfrac{n}{2}$

Cancelando 2 com 2:

$S_{n}=(2x+n)\cdot n\\\\\boxed{\boxed{S_{n}=2xn+n^{2}}}$