The ratio of the 4th term of a GP is 36 find the possible common ratio
Mathematics
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Step-by-step solution
5 stepsThe problem statement "The ratio of the 4th term of a GP is 36" is a bit unclear. We will assume it means "The ratio of the 4th term to the 2nd term of a GP is 36", as this interpretation leads to possible common ratios (plural), which the question asks for.
Recall the formula for the n-th term of a Geometric Progression (GP).
The n-th term of a GP is given by:
aₙ = a₁ * r^(n-1)
where a₁ is the first term and r is the common ratio.Write the expressions for the 4th term (a₄) and the 2nd term (a₂).
For n = 4:
a₄ = a₁ * r^(4-1)
a₄ = a₁ * r³For n = 2:
a₂ = a₁ * r^(2-1)
a₂ = a₁ * rSet up the equation based on the assumption that the ratio of the 4th term to the 2nd term is 36.
a₄ / a₂ = 36
Substitute the expressions for a₄ and a₂ into the equation and simplify.
(a₁ * r³) / (a₁ * r) = 36
r² = 36Solve for the common ratio, r.
r = ±√(36)
r = ±6The possible common ratios are 6 and -6.
The possible common ratios are 6 and -6.
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