Find the second term of a geometric sequence if

Question

a₃=18 and a₅=162

Answer 1

Use the explicit formula for a geometric sequence:

a_n = a₁(r^n-1)     Let a₁ = 18 and a₃ = 162

a₃ = a₁(r^3-1)

162 = 18r²

9 = r²

r = ±3

a₂ = 18/3 = 6  or  18/(-3) = -6 

 

Answer 2

For a geometric sequence, the nth term, a_n, is given by the formula a_n = a₁r^n-1, where a₁ is the first term and r is the common ratio.  

 

We have:  a₃ = a₁r² = 18

 

                a₅ = a₁r⁴ = 162

 

From the first equation, a₁ = 18/r².

 

Substituting into the second equation, we get 18r² = 162.  So, r² = 9.

 

Therefore, r = 3 and a₁ = 18/(3)² = 2

 

So, a₂ = a₁r = 6

 

Answer 3

Assuming a common ratio,

 

a₅/a₄ = a₄/a₃

 

a₄ = √(a₃a₅) = √[18(162)] = 54