Sequences: including Fibonacci-type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer and r is a rational number > 0)

The idea

This row covers three sequence families. A Fibonacci-type sequence forms each term by adding the two previous terms, so from 2, 5 the next term is 7 and then 12; the starting terms need not be 1 and 1. A quadratic sequence has a constant second difference, for example 3, 6, 11, 18, 27 with first differences 3, 5, 7, 9 and second difference 2. A simple geometric progression multiplies by a fixed rational ratio r greater than 0, written rⁿ for integer n, so 2, 4, 8, 16 has r = 2 and nth term 2ⁿ. You must recognise each type from its terms, continue it correctly and, where asked, find a stated term in the sequence.

What earns the marks

• Recognises a Fibonacci-type sequence as one built by adding the two preceding terms, and continues it correctly from any given starting pair.
• Recognises a quadratic sequence from a constant second difference and continues the sequence to find the next terms.
• Identifies a simple geometric progression by a constant multiplier r, where r is a rational number greater than 0, and continues it by repeated multiplication.
• Uses the form rⁿ with integer n to generate or check terms of a geometric progression, including fractional ratios such as r = 1/2.
• Distinguishes the three families when several are offered, for example rejecting an arithmetic sequence as Fibonacci-type or quadratic.
• Applies the correct rule to find a named term, such as the 10th term, rather than only the next term.

Common mistakes and how to put them right

• Treating any sequence with a pattern as Fibonacci-type.

  Put it right: A Fibonacci-type sequence specifically adds the two previous terms, so 4, 7, 10, 13 is arithmetic, not Fibonacci-type.

• Assuming every quadratic sequence has second difference 1.

  Put it right: The second difference can be any non-zero constant, such as 2 or 6.

• Using a constant difference for a geometric progression.

  Put it right: A geometric progression has a constant multiplier r, so 2, 4, 8, 16 multiplies by 2 each time.

• Allowing r to be zero or negative when the row restricts r to a rational number greater than 0.

  Put it right: Check that each term is obtained by multiplying by the same positive rational value.

Exam tips

• Write the first differences, and then the second differences, before deciding whether a sequence is arithmetic, quadratic or neither.
• For a geometric progression, divide a term by the previous term to confirm the multiplier r is constant before using rⁿ.
• Check a Fibonacci-type continuation by confirming each new term equals the sum of the two terms immediately before it.