Part 8 · Finding unknowns
Sequences with surds
Here is a sequence: \(\sqrt{3}\), 3, \(3\sqrt{3}\), 9, … Is there the same difference each time, or the same multiplier? Find the next term.
Working
Have a go first. Then open the working one line at a time.
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Same difference?
\(3 - \sqrt{3} \approx 1.27\), \(3\sqrt{3} - 3 \approx 2.20\)
The differences are not equal, so it is not arithmetic.
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Same multiplier?
\(\sqrt{3} \times \sqrt{3} = 3\), \(3 \times \sqrt{3} = 3\sqrt{3}\), \(3\sqrt{3} \times \sqrt{3} = 9\)
Each term is the one before times \(\sqrt{3}\), so it is geometric with ratio \(\sqrt{3}\).
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The next term
\(9 \times \sqrt{3} = 9\sqrt{3}\)
Multiply by \(\sqrt{3}\) again.