Question

Calcule a soma dos 18 primeiros termos da PG da foto acima.​

Answer 1

Explicação passo-a-passo:

• Razão

$\sf q=\dfrac{a_2}{a_1}$

$\sf q=\dfrac{6}{3\sqrt{2}}$

$\sf q=\dfrac{6}{3\sqrt{2}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}}$

$\sf q=\dfrac{6\sqrt{2}}{3\cdot2}$

$\sf q=\dfrac{6\sqrt{2}}{6}$

$\sf q=\sqrt{2}$

• Soma

A soma dos n primeiros termos de uma PG é dada por:

$\sf S_n=\dfrac{a_1\cdot(q^n-1)}{q-1}$

$\sf S_{18}=\dfrac{3\sqrt{2}\cdot[(\sqrt{2})^{18}-1)]}{\sqrt{2}-1}$

$\sf S_{18}=\dfrac{3\sqrt{2}\cdot(512-1)}{\sqrt{2}-1}$

$\sf S_{18}=\dfrac{3\sqrt{2}\cdot511}{\sqrt{2}-1}$

$\sf S_{18}=\dfrac{1533\sqrt{2}}{\sqrt{2}-1}$

$\sf S_{18}=\dfrac{1533\sqrt{2}}{\sqrt{2}-1}\cdot\dfrac{\sqrt{2}+1}{\sqrt{2}+1}$

$\sf S_{18}=\dfrac{1533\cdot2+1533\sqrt{2}}{(\sqrt{2})^2-1^2}$

$\sf S_{18}=\dfrac{3066+1533\sqrt{2}}{2-1}$

$\sf S_{18}=\dfrac{3066+1533\sqrt{2}}{1}$

$\sf S_{18}=3066+1533\sqrt{2}$

$\sf \red{S_{18}=1533\cdot(2+\sqrt{2})}$

Answer 2

Vamos lá :

q  = a₂/a₁  = 6/3√2  = 2/√2 * √2/√2  = 2√2/(√2)²  = 2√2/2  = √2

Sn  = a₁ . (qⁿ - 1)/(q - 1)

S₁₈  = 3√2 . ((√2)¹⁸ - 1)/(√2 - 1)

S₁₈  = 3√2 .(2⁹ - 1)/(√2 - 1)

S₁₈  = 3√2 . (512 - 1)/(√2 - 1)

S₁₈  = 3√2 . 511/(√2 - 1)

S₁₈  = 1533√2/(√2 - 1)   * (√2 + 1)/(√2 + 1)

S₁₈  = 1533√2 .(√2 + 1)/((√2)² - 1²)

S₁₈  = 1533.(√2)² + 1533√2/(2 -1)

S₁₈  = 1533.2 + 1533√2/1

S₁₈  = 3066 + 1533√2

Espero ter ajudado !!!