If G₃ = 16/3 and G₅ = 64/27, then find G₁ and the common ratio.

Given that G₃ = 16/3 and G₅ = 64/27, we can find G₁ and the common ratio by using the formula for the geometric progression:

G₃ = G₁ * r²
G₅ = G₁ * r⁴

Substitute the given values into the equations:

16/3 = G₁ * r²
64/27 = G₁ * r⁴

Now, solve for G₁ and r by solving the system of equations:

From the first equation:
G₁ = (16/3) / r²

Substitute G₁ into the second equation:
64/27 = ((16/3) / r²) * r⁴
64/27 = 16r²
r² = 4/27
r = 2/3 or -2/3

Now, substitute r back into the equation to find G₁:

G₁ = (16/3) / (2/3)²
G₁ = (16/3) / (4/9)
G₁ = (16/3) * (9/4)
G₁ = 36/2
G₁ = 18

Therefore, G₁ = 18 and the common ratio can be either 2/3 or -2/3.

Step 1: Find the common ratio (r)

Divide G₅ by G₃ to get the common ratio:

r = G₅/G₃ = (64/27) / (16/3) = 4/9

Step 2: Find G₁

Since G₁ is the first term in the sequence, we can use the formula:

G₁ = G₃/r^2

Substituting the values we have:

G₁ = 16/3 / (4/9)^2 = 16/3 / 16/81 = 16/3 - 81/16 = 81/3 = 27

Therefore, G₁ = 27 and the common ratio is 4/9.