Sequence terms and limiting values

A limiting value describes what the terms approach as the index increases.

To calculate a specified term, substitute its positive integer index into the nth-term expression. For a rational expression, divide numerator and denominator by the highest power of n present in the denominator. Terms such as 1/n tend to zero. Their signs still determine whether finite terms lie above or below the limit.

A sequence can approach a value without ever equalling it. Solving for an index is a separate task: the index must be a positive integer. An algebraic solution n = 7/2 therefore does not identify a term of the sequence. Do not replace a finite-term difference with a difference of limits.

Worked example

For uₙ = (3n + 2)/(n + 4), find u₂ and the limit as n increases.

u₂ = (6 + 2)/(2 + 4) = 4/3.
Divide top and bottom by n: uₙ = (3 + 2/n)/(1 + 4/n).
Both fractions containing 1/n tend to zero.

Answer: u₂ = 4/3; limiting value 3

Practise sequence terms and limiting values

Course mapping

These specification references show where the topic occurs. The questions cover only some parts of each topic.

8365 · core · 2.20: Sequence terms and limiting values
8365 · core · 2.20: A rational sequence limiting value

Next practice: Quadratic sequences.