Lesson Explainer: The General Term of a Sequence Mathematics

In this explainer, we will learn how to use the general term or a recursive formula of a sequence to work out terms in the sequence.

We begin by recalling that a sequence is an ordered list of numbers, each of which is called a term for example, 2,4,6,8,10,… and 1,2,4,8,16,….

When dealing with sequences, we can usually find the next term by spotting a general rule or pattern.

In the examples above, this can be done by adding 2 and multiplying by 2 respectively.

Each term in a sequence is derived from a particular rule that relates to its position in the sequence or that relates each term to the term before it.

The notations 𝑇,𝑇,𝑇,…,𝑇,𝑇 are used to denote the individual terms in a sequence. The expression given by 𝑇 is known as the general term, or the 𝑛th term, of the sequence.

Definition: General Term of a Sequence

The general term of a sequence, sometimes called the 𝑛th term and written as 𝑇, is an algebraic expression that relates the term to its position number in the sequence.

Consider the general term 𝑇=3𝑛+4.

Since this is the 𝑛th term of the sequence, it follows that to find the 8th term, we would substitute 𝑛=8 into the given formula: 𝑇=3×8+4=28.

Similarly, to find the first three terms, we would begin by finding the first term by letting 𝑛=1 as follows: 𝑇=3×1+4=7.

The second term is given by 𝑛=2 as follows: 𝑇=3×2+4=10.

In a similar way, the third term is found by substituting 𝑛=3 as follows: 𝑇=3×3+4=13.

Hence, the first three terms are 7, 10, and 13.

Let us summarize this as follows.

How To: Using the General Term of a Sequence

If the general term of a sequence contains an expression in 𝑛, substitute the term number in place of 𝑛 to find a specific term in the sequence.

In our first two examples, we will demonstrate how to calculate the first five terms of a sequence given its general term.

Example 1: Finding Terms of a Sequence given Its General Term

Find the first five terms of the sequence whose 𝑛th term is given by 𝑇=𝑛−14, where 𝑛≥1.

Answer

To calculate the first five terms of the sequence, we substitute 𝑛=1,2,3,4, and 5 into the formula 𝑇=𝑛−14 as follows: 𝑇=(1)−14=−13,𝑇=(2)−14=−10,𝑇=(3)−14=−5,𝑇=(4)−14=2,𝑇=(5)−14=11.

Therefore, the first five terms of the sequence are −13, −10, −5, 2, and 11.

Example 2: Finding Terms of a Sequence given Its General Term

Find the first five terms of the sequence whose 𝑛th term is given by 𝑇=5𝑛+𝑛.

Answer

To calculate the first five terms of the sequence, we substitute 𝑛=1,2,3,4, and 5 into the formula 𝑇=5𝑛+𝑛 as follows: 𝑇=5(1)+(1)=5+1=6,𝑇=5(2)+(2)=20+8=28,𝑇=5(3)+(3)=45+27=72,𝑇=5(4)+(4)=80+64=144,𝑇=5(5)+(5)=125+125=250.

Therefore, the first five terms of the sequence are 6, 28, 72, 144, and 250.

In our next example, we need to calculate a specific term of a sequence given its general term.

Example 3: Finding a Specified Term of a Sequence given Its General Term

Find the seventh term of the sequence 𝑇=𝑛−14.

Answer

To calculate the seventh term of the sequence, we substitute 𝑛=7 into the formula 𝑇=𝑛−14 as follows: 𝑇=(7)−14=343−14=329.

Therefore, the seventh term of the sequence is 329.

Before looking at our next example, we will consider what it means to have a recursive formula as the general term. This occurs when the general term is an algebraic expression that relates to the term that precedes it.

Definition: Recursive Formula

A sequence can be defined by giving one general term of the sequence as an expression using other terms of the sequence. This relationship between terms occurs throughout the sequence and is therefore called a recurrence relation.

Consider the general term 𝑇=2𝑇+5, where 𝑛≥2, and 𝑇=4.

We can see that this formula contains an expression for 𝑇. This is the term before 𝑇, so this rule tells us that to find a term in the sequence, we multiply the preceding term by 2 and then add 5.

We can calculate 𝑇 and 𝑇 as follows:𝑇=2𝑇+5𝑇=2×4+5=13,𝑇=2𝑇+5𝑇=2×13+5=31.

Hence, the first three terms are 4, 13, and 31.

Let us summarize this as follows.

How To: Using the Recursive Formula for a Sequence

If the general term of a sequence contains an expression in 𝑇, substitute the preceding term in place of 𝑇 to find any term.

If the general term contains an expression for 𝑇 in 𝑇, substitute the value of the preceding term in place of 𝑇 to find the value of 𝑇.

Another example of a recursive formula is the formula used to describe the Fibonacci sequence: 1,1,2,3,5,8,13,21,34,….

Each term of the Fibonacci sequence is related to the terms that precede it. The Fibonacci sequence cannot easily be described using a formula linking the terms with their position number. Instead, we describe the sequence using a recursive formula.

The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms: 𝑇=1,𝑇=1,𝑇=𝑇+𝑇,𝑛>2.for

To find the 10th term of the sequence, for example, we would need to add the 8th and 9th terms as follows: 𝑇=𝑇+𝑇𝑇=34+21=55.

Having demonstrated how a recursive formula works, let us consider a more complicated example involving variable exponents.

Example 4: Finding Terms of a Sequence given a Recursive Formula

Find the first five terms of the sequence 𝑇, given 𝑇=(−1)9𝑇, 𝑛≥1, and 𝑇=−11.

Answer

We begin by recognizing that the formula 𝑇=(−1)9𝑇 is an example of a recursive formula, where 𝑇 is the term after 𝑇.

In order to calculate consecutive terms of the sequence, we need to substitute the preceding term in place of 𝑇.

Since 𝑇=−11, 𝑇=(−1)9𝑇=(−1)9(−11)=−1−99=199.

We now substitute this value of 𝑇 into the formula such that 𝑇=(−1)9𝑇=(−1)9=1=11.

Next, we substitute the value of 𝑇 into the formula to calculate 𝑇 as follows: 𝑇=(−1)9𝑇=(−1)9(11)=−199.

Finally, we substitute the value of 𝑇 into the formula to calculate 𝑇 as follows: 𝑇=(−1)9𝑇=(−1)9=1=−11.

Therefore, the first five terms of the sequence are −11, 199, 11, −199, and −11.

It’s interesting to note that, in the previous example, the first term, 𝑇, was equal to the fifth term, 𝑇. This means that we had a periodic sequence where the first four terms were repeated.

This can be illustrated as follows: 𝑇=𝑇=𝑇…,𝑇=𝑇=𝑇…,𝑇=𝑇=𝑇…,𝑇=𝑇=𝑇….

In our final example, we will identify which of the given formulae satisfy the given sequence.

Example 5: Verifying the General Term of a Sequence

Which of the following is the formula for the general term of the sequence 52, 84, 116, 148?

  1. 52+30(𝑛−1)
  2. 52+32(𝑛−1)
  3. 52+32(𝑛+1)
  4. 84+32(𝑛−1)
  5. 84+30(𝑛+1)

Answer

Since the first four terms of the sequence are 52, 84, 116, and 148, one way of working out the correct formula for the general term of the sequence 52, 84, 116, 148 is to substitute 𝑛=1,2,3, and 4 into each of the options.

Let us consider option (A) such that 𝑇=52+30(𝑛−1).

This gives us 𝑇=52+30(1−1)=52,𝑇=52+30(2−1)=82,𝑇=52+30(3−1)=112,𝑇=52+30(4−1)=142.

The formula 52+30(𝑛−1) gives us the sequence 52, 82, 112, 142, so this is not the correct answer.

Let us consider option (B) such that 𝑇=52+32(𝑛−1).

This gives us 𝑇=52+32(1−1)=52,𝑇=52+32(2−1)=84,𝑇=52+32(3−1)=116,𝑇=52+32(4−1)=148.

The formula 52+32(𝑛−1) gives us the sequence 52, 84, 116, 148, so this is the correct answer.

While we have found that option (B) is correct, it is worth calculating the first four terms of the other three formulae to confirm that they are not correct.

Let us consider option (C) such that 𝑇=52+32(𝑛+1).

This gives us 𝑇=52+32(1+1)=116,𝑇=52+32(2+1)=148,𝑇=52+32(3+1)=180,𝑇=52+32(4+1)=212.

The formula 52+32(𝑛+1) gives us the sequence 116, 148, 180, 212, so this is not the correct answer.

Let us consider option (D) such that 𝑇=84+32(𝑛−1).

This gives us 𝑇=84+32(1−1)=84,𝑇=84+32(2−1)=116,𝑇=84+32(3−1)=148,𝑇=84+32(4−1)=180.

The formula 84+32(𝑛−1) gives us the sequence 84, 116, 148, 180, so this is not the correct answer.

Let us consider option (E) such that 𝑇=84+30(𝑛+1).

This gives us 𝑇=84+30(1+1)=144,𝑇=84+30(2+1)=174,𝑇=84+30(3+1)=204,𝑇=84+30(4+1)=234.

The formula 84+30(𝑛+1) gives us the sequence 144, 174, 204, 234, so this is not the correct answer.

We can therefore confirm that options (A), (C), (D), and (E) are incorrect. The correct formula is 52+32(𝑛−1).

We will finish this explainer by recapping some of the key points.

Key Points

  • The general term of a sequence, sometimes called the 𝑛th term and written 𝑇, is an algebraic expression that relates the term to its position number or the term that precedes it (this is called the recursive formula).
  • If the general term of a sequence contains an expression in 𝑛, substitute the term number in place of 𝑛.
  • If the general term of a sequence contains an expression in 𝑇, substitute the preceding term in place of 𝑇.
  • If the general term contains an expression for 𝑇 in 𝑇, substitute the value of the preceding term in place of 𝑇 to find the value of 𝑇.

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