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Geometric Progression Problem: Finding the First Term Given Sum and Difference Conditions
A Classic Geometric Progression Problem to Warm Up
Let’s take a break and stretch our math muscles. Since we’ve been talking about geometric progressions, here’s a problem worth thinking through:
Consider 4 consecutive terms of a geometric progression a < b < c < d, where two conditions are known:
- a + b + c + d = 1320
- c − a = 264
Find the first term, a.
Setting Up the Geometric Sequence
The standard approach is to express all four terms using the first term and the common ratio. Write each term as a product of the first term and the appropriate power of the common ratio:
- a = a
- b = aq
- c = aq²
- d = aq³
This gives a system of two equations in two unknowns, a and q.
Solving the Geometric Progression Problem
Substituting into the two conditions:
- a + aq + aq² + aq³ = 1320
- aq² − a = 264
From the second equation, factor out a: a(q² − 1) = 264, so a = 264 / (q² − 1).
Substitute into the first equation and simplify to find the value of q. This yields two valid solutions: q = 2 and q = 3.
Answer: Two Solutions
Plugging each value of q back in to find a:
- When q = 2: a = 88
- When q = 3: a = 33
Both solutions satisfy the original conditions, so the answer to this geometric sequence olympiad problem is a = 33 or a = 88.
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