Question

What is the first term of the geometric sequence presented in the table below?

n 4 9
an 6 −192

Hint: an = a1(r)n − 1, where a1 is the first term and r is the common ratio.

Answer 1

The equation for the n-th term of the geometric sequence is

`a_(n) = a_(1)r^(n-1)`
where
a₁ = the first term
r = the common ratio.

a₄ = 6, therefore
a₁ r³ = 6 (1)

a₉ = -192, therefore
a₁ r⁸ = -192 (2)

Divide equation (2) by equation (1).

`(a_(1)r^(8))/(a_(1)r^(3) ) = (-192)/(6) \\ r^(5) = -32 = (-2)^(5) \\ r=-2`

From (1), obtain
a₁(-2)³ = 6
-8a₁ = 6
a₁ = -3/4

Answer: -3/4