In a certain geometric sequence, the second term is 12 and the fifth term is 96. What is the common ratio of the sequence? - Coaching Toolbox
In a certain geometric sequence, the second term is 12 and the fifth term is 96. What is the common ratio of the sequence?
In a certain geometric sequence, the second term is 12 and the fifth term is 96. What is the common ratio of the sequence?
When users encounter intriguing number patterns online—especially in math or finance—geometric sequences often spark both fascination and curiosity. A common question arises: if the second term is 12 and the fifth term is 96, what’s the consistent amount by which each step multiplies? This pattern reveals hidden math behind trends, income calculations, and long-term forecasts—making it more than just academic. Understanding the common ratio helps decode growth, compounding, and predictable change in real-life scenarios.
Why Is This Sequence Trending in User Conversations?
Understanding the Context
In a digital age increasingly focused on data literacy, clean math and pattern recognition have become more valuable than ever. People search for clear explanations to build confidence in financial planning, investment analysis, or even productivity modeling—where growth rates matter more than raw numbers. This specific sequence often surfaces in discussions around compound returns, scaling business models, or predicting stepwise increases, echoing real-world patterns where small inputs lead to exponential outcomes. Its presence in search queries reflects a growing interest in accessible mathematical reasoning tied to everyday decision-making.
Breaking Down How to Find the Common Ratio
In a geometric sequence, each term is found by multiplying the previous one by a fixed value called the common ratio—usually denoted as r. From the given:
- The second term: ( a_2 = 12 )
- The fifth term: ( a_5 = 96 )
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The formula for the nth term is ( a_n = a_1 \cdot r^{n-1} ), so:
- ( a_2 = a_1 \cdot r = 12 )
- ( a_5 = a_1 \cdot r^4 = 96 )
Dividing the fifth term by the second term eliminates ( a_1 ) and isolates the ratio:
[ \frac{a_5}{a_2} = \frac{a_1 \cdot r^4}{a_1 \cdot r} = r^3 ]
So:
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[ r^3 = \frac{96}{12} = 8 ]
Taking the cube root gives:
[ r = \sqrt[3]{8} = 2 ]
This confirms the common ratio is 2—each term doubles to reach the next, illustrating a steady geometric progression.
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