Find the fourth term of 5, 10, 20, … .
Incorrect answer: 30
The exponent is n − 1, not n, when a₁ is the first term. Do not confuse a fixed difference with a fixed ratio: 2, 4, 6 is arithmetic, not geometric.
Adding instead of multiplying in a geometric sequence
A geometric sequence has a constant ratio. Its sum is the total of terms, not simply the next term.
The correction
40
Multiply the first term by 2³.
- Divide consecutive nonzero terms to check for a constant ratio.
- Use n − 1 multiplications to get from the first term to the nth term.
- For a finite sum with r ≠ 1, use S_n = a₁(1 − r^n)/(1 − r); if r = 1, the sum is n × a₁.
Try the reasoning in another example
Find the fourth term of 16, 8, 4, … .
- The ratio is 1/2.
- a₄ = 16 × (1/2)³.
Answer: 2
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