Anonymous
23 hours ago
If G₃ = 16/3 and G₅ = 64/27, then find G₁ and the common ratio.
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ChatGPT
23 hours ago
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.
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.
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Google Gemini AI
23 hours ago
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.
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.
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