Generate a geometric sequence with a surd ratio

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Generate a geometric sequence with a surd ratio

A positive surd can be the common ratio of a geometric sequence. With first term 1 and ratio √2, the terms are 1, √2, 2, 2√2, 4. Exact surd forms retain the multiplicative structure without rounding drift.

The position rule is (√2)ⁿ⁻¹ when the first position is 1. Even exponents produce integer powers of 2; odd exponents retain a factor of √2. The laws of indices explain this pattern rather than merely describing its appearance.

Keep exact values while generating later terms. Rounding √2 before repeated multiplication can accumulate error, especially when checking whether a particular term has an exact integer value.

Worked example: Find the fifth term of the stated sequence.

Use four factors of √2.

Pair the factors into two groups of 2.

Answer: 4.

Check your understanding: What is the sixth term?

Answer: 4√2.

Why: Multiply the fifth term by the same ratio.