Question

Determine the

Answer 1

Answer:

a₄ = - $\frac{27}{2}$

Step-by-step explanation:

In a geometric sequence, the common ratio r is calculated as

r = $\frac{a_{2} }{a_{1} }$ = $\frac{a_{3} }{a_{2} }$ , substitute values

$\frac{2x+2}{x}$ = $\frac{3x+3}{2x+2}$ ( cross- multiply )

(2x + 2)² = x(3x + 3) ← expand both sides

4x² + 8x + 4 = 3x² + 3x ( subtract 3x² + 3x from both sides )

x² + 5x + 4 = 0 ← in standard form

(x + 1)(x + 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 1 = 0 ⇒ x = - 1

x + 4 = 0 ⇒ x = - 4

However x ≠ - 1 as this would make the denominator of the second term equal to zero and therefore undefined.

Thus x = - 4 and

r = $\frac{2(-4)+2}{-4}$ = $\frac{-8+2}{-4}$ = $\frac{-6}{-4}$ = $\frac{3}{2}$

Then

a₁ = - 4

a₂ = - 4 × $\frac{3}{2}$ = - 6

a₃ = - 6 × $\frac{3}{2}$ = - 9

a₄ = - 9 × $\frac{3}{2}$ = - $\frac{27}{2}$