Geometric progression members

We get three consecutive GP members if we subtract the same number from 33, 45, and 63. Determine this GP and calculate its fifth member.

Final Answer:

a₁ = 24
q = 1.5
a₅ = 121.5

b_{1} = 33 b_{2} = 45 b_{3} = 63 a_{1} = b_{1} - d a_{2} = b_{2} - d = q \cdot a_{1} a_{3} = b_{3} - d = q^{2} \cdot a_{1} b_{2} - d = q \cdot (b_{1} - d) b_{3} - d = q^{2} \cdot (b_{1} - d) q = (b_{2} - d) / (b_{1} - d) b_{3} - d = (b_{2} - d) / (b_{1} - d) \cdot (b_{2} - d) / (b_{1} - d) \cdot (b_{1} - d) (b_{3} - d) \cdot (b_{1} - d) = (b_{2} - d) \cdot (b_{2} - d) (b_{3} - d) \cdot (b_{1} - d) = (b_{2} - d) \cdot (b_{2} - d) - 6 d + 54 = 0 - 6 \cdot d + 54 = 0 6 d = 54 d = \frac{5 4}{6} = 9 d = 9 q = (b_{2} - d) / (b_{1} - d) = (45 - 9) / (33 - 9) = \frac{3}{2} = 1 \frac{1}{2} = 1.5 a_{1} = b_{1} - d = 33 - 9 = 24

q = 1.5 = \frac{3}{2} = 1 \frac{1}{2} Verifying Solution: a_{2} = b_{2} - d = 45 - 9 = 36 a_{3} = b_{3} - d = 63 - 9 = 54 A_{1} = a_{1} = 24 A_{2} = q \cdot A_{1} = 1.5 \cdot 24 = 36 A_{3} = q \cdot A_{2} = 1.5 \cdot 36 = 54

a_{5} = a_{1} \cdot q^{4} = 24 \cdot 1 . 5^{4} = 121.5

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