Teach Me Maths

Sequences - nth Terms and Limiting Values

How to use and find nth terms, identify term positions, handle quadratic sequences and work out limiting values.

Distinguish a term number from a term value

The first term has n=1, unless another starting index is stated. A term formula gives the value when you substitute its position. To find a position, set the formula equal to the desired value and solve for a positive integer n

For an arithmetic sequence with first term a and common difference d:

u_n=a+(n-1)d

There are n-1 steps from the first term to the nth term.

Worked example 1

A sequence has u_n=4n-9. Find its first three terms and its 25th term.

  1. Use n=1,2,3: the first terms are -5,-1,3

  2. Your turn. Calculate u_{25}=4(25)-9

    u_{25}=91

  3. Answer: first terms -5,-1,3; 25th term 91. The subscript is the position, not a multiplier.

Worked example 2

The sequence starts 17,12,7,2,\dots. Find its nth term and the position of -83

  1. The common difference is -5, so u_n=17-5(n-1)

  2. Your turn. Simplify the nth-term expression.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    u_n=22-5n

  3. Set 22-5n=-83. Then -5n=-105, so n=21
  4. Answer: u_n=22-5n, and -83 is the 21st term.

Worked example 3

An arithmetic sequence starts r+2s, r+5s, r+8s, \dots. Find its nth term.

  1. Your turn. What is its common difference?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The common difference is 3s

  2. Use first term plus (n-1) differences: u_n=r+2s+3s(n-1)
  3. Answer: u_n=r+(3n-1)s. At n=1, this gives r+2s as required.

Use second differences for a quadratic sequence

If u_n=an^2+bn+c, the second difference is 2a. Find a, subtract the sequence an^2, then find the linear rule for what remains. Check more than the first term.

Worked example 4

Find the nth term of 6,15,28,45,66,\dots

  1. The first differences are 9,13,17,21; the second differences are all 4

  2. Your turn. If 2a=4, what is a?

    a=2

  3. Subtract 2n^2: the remainders are 4,7,10,13,16, whose nth term is 3n+1
  4. Answer: u_n=2n^2+3n+1. For example, u_3=18+9+1=28

Worked example 5

A sequence has u_n=-n^2+11n-18. Which terms have value zero?

  1. Set -n^2+11n-18=0, then multiply by -1: n^2-11n+18=0

  2. Your turn. Solve (n-2)(n-9)=0

    Give every value, separated by commas

    n=2 or n=9

  3. Answer: the 2nd and 9th terms. Both positions are positive integers; reject any non-integer or non-positive position in this type of question.

Find the value approached as the position grows

A limiting value is the number the terms approach as n\to\infty. For a ratio of linear expressions, divide the numerator and denominator by n. Terms such as 1/n approach zero; they are not zero at any finite positive n

Not every sequence has a finite limiting value. For example, a sequence that increases without bound does not approach a fixed real number.

Worked example 6

For u_n=\frac{5n-2}{2n+3}, find u_{20} and the limiting value.

  1. u_{20}=\frac{100-2}{40+3}=\frac{98}{43}. To find the limit, rewrite u_n=\frac{5-2/n}{2+3/n}

  2. Your turn. As n\to\infty, what value does this ratio approach?

    The numerator approaches 5 and the denominator approaches 2

  3. Answer: u_{20}=\frac{98}{43}; limiting value \frac52. A particular term and the limiting value are different quantities.

Explore terms approaching a limit

Try it: compare terms 10, 30 and 60. The distance to the limit is 6/(n+2) and remains positive. A sequence plot uses separate dots because its positions are integers.

Worked example 7

For u_n=4+\frac6{n+2}, find the limit and the first term whose distance from it is less than 0.1

  1. The fraction approaches zero, so the limit is 4. The distance from 4 is \frac6{n+2}
  2. Require \frac6{n+2}<\frac1{10}. Since n+2>0, multiply to get 60<n+2, so n>58

  3. Your turn. What is the first allowed integer position?

    The first position is n=59

  4. Answer: limit 4; the 59th term is the first within the stated distance. At n=58 the distance is exactly 0.1, so it is excluded.

Common mistakes

  • Starting at the wrong position. Use n=1 for the first term unless told otherwise.
  • Confusing a term’s value with its position. n is the position; u_n is the value at that position.
  • Using the second difference as the leading coefficient. For a quadratic sequence, halve it to get the coefficient of n^2.
  • Accepting a fractional term position. A term position must be an integer in the allowed range.
  • Assuming a limit must be reached. Check the term formula; approaching a value does not guarantee equality.

Now try it: Sequences - nth Terms and Limiting Values practice questions

More on this topic: Sequences - nth Terms and Limiting Values worksheet with full solutions