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MathidiaMathematics Academy
Sequences and iteration
Lesson 30 of 75
Review

Sequence reasoning

Objective. Prove divisibility and membership statements about sequences.

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  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Sequence reasoning uses algebraic expressions for the nth term to prove general statements, such as divisibility or whether a given number belongs to the sequence. To prove every term is divisible by a number k, write un in a form that clearly shows a factor of k, for example un = 3n² + 6n = 3n(n+2), which is always divisible by 3. To check whether a value belongs to a sequence, set un equal to that value and solve for n; the value is a term only if n comes out as a positive whole number. For un = 2n + 5, testing un = 47 gives n = 21, a valid term, but un = 48 gives n = 21.5, not a term. Always justify divisibility by showing an explicit common factor algebraically.

Worked example

Prove that every term of the sequence un = 2n² + 4n is divisible by 2.

2 … 4
Illustrated card showing 2 … 4
  1. 1Write un = 2n² + 4n
  2. 2Factor out 2: un = 2(n² + 2n)
  3. 3Since n² + 2n is always an integer when n is a positive integer, 2(n² + 2n) is a multiple of 2
  4. 4Therefore un is divisible by 2 for every positive integer n

Answer: un = 2(n² + 2n), always a multiple of 2

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Common mistake. Pupils claim a number is in the sequence just because it is close to another term, without solving un = value for a whole-number n. For un = 4n − 1, they might assume 50 is a term because 49 is, but solving gives n = 12.75, so 50 is not a term. Always solve algebraically and check n is a positive integer.

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