Further Sequences [H]
Further Sequences
At Higher tier, geometric sequences extend to cases where the common ratio \(r\) is a surd (an irrational square root) or a negative rational number. The same nth term formula \( T_n = a cdot r^{n-1} \) applies, but care is needed when evaluating surds and when the sign alternates. Other sequences of increasing complexity may also arise — including combinations of polynomial and exponential terms.
Geometric Sequences with Surd Ratio
When \( r = \sqrt{k} \), the terms involve increasing powers of \(\sqrt{k}\) and can be simplified using surd rules: \( (\sqrt{k})^2 = k \), \( (\sqrt{k})^3 = k\sqrt{k} \), etc.
Example: \( a=1, r=\sqrt{2} \). Terms: \( 1, \sqrt{2}, 2, 2\sqrt{2}, 4, 4\sqrt{2}, 8, ldots \) — alternating between surds and integers.
Geometric Sequences with Negative Ratio
When \( r < 0 \), the sequence alternates in sign. Each term changes sign from the previous one, while growing or decaying in magnitude depending on \( |r| \).
Recognising Complex Sequence Types
Some sequences require multiple checks. Work through: constant first differences (arithmetic) → constant second differences (quadratic) → constant ratio (geometric) → other (Fibonacci-type, combined, etc.).
Worked Examples
A geometric sequence has first term \( 2 \) and common ratio \( \sqrt{3} \). Write the first five terms in simplified surd form.
\[ T_1=2,quad T_2=2\sqrt{3},quad T_3=2(\sqrt{3})^2=6,quad T_4=6\sqrt{3},quad T_5=6(\sqrt{3})^2=18 \]A sequence is \( 1, \sqrt{5}, 5, 5\sqrt{5}, ldots \) Identify it as geometric and state the ratio.
Each term is multiplied by \( \sqrt{5} \): ratio \( r = \sqrt{5} \). nth term: \( (\sqrt{5})^{n-1} = 5^{(n-1)/2} \).
A geometric sequence has first term 6 and ratio \( -\frac{1}{2} \). Find the first five terms and the 8th term.
\[ 6, -3, \frac{3}{2}, -\frac{3}{4}, \frac{3}{8} \] \[ T_8 = 6 \times \left(-\frac{1}{2}\right)^7 = 6 \times \left(-\frac{1}{128}\right) = -\frac{6}{128} = -\frac{3}{64} \]Classify each: (i) 3, 6, 12, 24 (ii) 2, 5, 10, 17, 26 (iii) 100, 10, 1, 0.1 (iv) 1, 1, 2, 3, 5, 8.
(i) Ratios: 2, 2, 2 — geometric, \(r=2\).
(ii) First diffs: 3, 5, 7, 9. Second diffs: 2, 2, 2 — quadratic.
(iii) Ratios: 0.1, 0.1, 0.1 — geometric, \(r=0.1\).
(iv) Each term = sum of two before — Fibonacci-type (the classic Fibonacci sequence).
Key Takeaways
- Geometric sequences with surd ratio: use surd laws to simplify terms. \( (\sqrt{k})^2=k \), \( (\sqrt{k})^3=k\sqrt{k} \), etc.
- Negative common ratio: the sequence alternates in sign; \( (-1)^{n-1} \) controls the sign pattern.
- Classification order: check first differences (arithmetic), then second differences (quadratic), then ratios (geometric), then Fibonacci-type.
- The nth term formula \( T_n=ar^{n-1} \) applies regardless of whether \(r\) is integer, fraction, surd, or negative.
- For a surd ratio \( r=\sqrt{k} \): even-numbered terms are rational (multiples of \(k\)); odd-numbered terms involve \(\sqrt{k}\).