Resposta:

Explicação passo-a-passo:

Letra a)

(x + 1)² + (x + 1)²/2 + (x + 1)²/4 + ... = 8 <=>

(x + 1)²[1 + 1/2 + 1/4 + 1/8 + ...] = 8 <=>

(x + 1)²[1/(1 - 1/2)] = 8 <=>

(x + 1)²[1/(1/2)] = 8 <=>

(x + 1)²[2] = 8 <=>

(x + 1)² = 8/2 <=>

(x + 1)² = 4 <=>

|x + 1| = 2 =>

x + 1 = 2 => x = 2 - 1 => x = 1

ou

x + 1 = - 2 => x = - 2 - 1 => x = - 3

S = {- 3, 1}

Letra b)

(x - 1)² + (x - 1)²/3 + (x - 1)²/9 + ... = 3/2 <=>

(x - 1)²[1 + 1/3 + 1/9 + ...] = 3/2 <=>

(x - 1)²[1/(1 - 1/3)] = 3/2 <=>

(x - 1)²[1/(3/3 - 1/3)] = 3/2 <=>

(x - 1)²[1/(3 - 1)/3] = 3/2 <=>

(x - 1)²[1/(2/3)] = 3/2 <=>

(x - 1)²[3/2] = 3/2 <=>

(x - 1)² = (3/2)/(3/2) <=>

(x - 1)² = 1 <=>

|x - 1| = 1 =>

x - 1 = 1 => x = 1 + 1 => x = 2

ou

x - 1 = - 1 => x = - 1 + 1 => x = 0

S = {0, 2}

Abraços!