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Algebra 1

4.18.3 Define an Arithmetic Sequence by the nth Term

Algebra 14.18.3 Define an Arithmetic Sequence by the nth Term

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Activity

1. Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again . . .

a. Instead of writing a recursive definition, Clare writes C ( n ) = 80 · ( 1 2 ) n C ( n ) = 80 · ( 1 2 ) n , where C C is the area, in square inches, of the paper after n n cuts. Explain where the different terms in her expression came from.

b. Approximately what is the area of the paper after 10 cuts?

2. Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away 1 inch of the width. Then he does it again, and again . . .

a. Complete the table for the area of Kiran’s paper K ( n ) K ( n ) , in square inches, after n n cuts.

n n K ( n ) K ( n )

0

80

1

2

80 8 8 = 80 8 ( 2 ) = 64 80 8 8 = 80 8 ( 2 ) = 64

3

4

5

b. Kiran says the area after 6 cuts, in square inches, is 80 8 · 6 80 8 · 6 . Explain where the different terms in his expression came from.

c. Write a definition for K ( n ) K ( n ) that is not recursive.

3. Which is larger, K ( 6 ) K ( 6 ) or C ( 6 ) C ( 6 ) ?

4. Is K ( n ) K ( n ) arithmetic or geometric?

5. Is C ( n ) C ( n ) arithmetic or geometric?

A recursive sequence is a sequence in which terms are defined using one or more of the previous terms. If you know the nth term of an arithmetic or geometric sequence and the common difference or factor, you can find the (n+1)th term by using the recursive formula.

Different definitions can often create the same sequence. For arithmetic and geometric sequences, there are general rules, called explicit rules, that can be followed to help find any term in the sequences.

These are called the nth  term or the general term of the sequence.

Arithmetic Sequence Formulas

Recursive Formula

a n = a n 1 + d a n = a n 1 + d , a 1 a 1 = first term

where a 1 a 1 is the first term, n n is the term you want, and d d is the common difference.

Explicit General Formula

a n = a 1 + ( n 1 ) d a n = a 1 + ( n 1 ) d

Where a 1 a 1 is the first term, n n is the term you want, and d d is the common difference.

Self Check

Self Check

What is the 12th term in an arithmetic sequence where the first term is 22 and the common difference is –2?

  1. 45056
  2. 0
  3. 22
  4. 2

Additional Resources

The nth Term of Arithmetic Sequences

GENERAL TERM (NTH TERM) OF AN ARITHMETIC SEQUENCE

The general term of an arithmetic sequence with first term a 1 a 1 and common difference d d is: a n = a 1 + ( n 1 ) d a n = a 1 + ( n 1 ) d .

Example 1

Write the explicit formula for the sequence given by the terms 18, 21, 24, 27, ...

Solution

Step 1- Write the general formula.

a n = a 1 + ( n 1 ) d a n = a 1 + ( n 1 ) d

Step 2 - Substitute values for the first term, common difference/ratio, term number.

a 1 = 18 a 1 = 18 , d = 3 d = 3

a n = 18 + ( n 1 ) 3 a n = 18 + ( n 1 ) 3

So, the explicit formula for the sequence 18, 21, 24, 27… is a n = 18 + ( n 1 ) 3 a n = 18 + ( n 1 ) 3 .

Now any term can more easily be determined - even the 1000th term!

Example 2

Find the 15th term of a sequence where the first term is 3 and the common difference is 6.

Step 1 - Write the general formula.

a n = a 1 + ( n 1 ) d a n = a 1 + ( n 1 ) d

Step 2 - Substitute values for the first term, common difference/ratio, term number.

a 1 = 3 , d = 6 a 1 = 3 , d = 6 , n = 15 n = 15

a 15 = 3 + ( 15 1 ) 6 a 15 = 3 + ( 15 1 ) 6

Step 3 - Simplify the expression.

a 15 = 3 + ( 14 ) 6 a 15 = 3 + ( 14 ) 6

a 15 = 3 + 84 a 15 = 3 + 84

a 1 5 = 87 a 1 5 = 87

If the explicit formula for this question was needed, we would not have substituted n = 15 n = 15 and the n n th term formula would have been a n = 3 + ( n 1 ) 6 a n = 3 + ( n 1 ) 6 . This formula is also equivalent to

a n = 3 + 6 ( n 1 ) a n = 3 + 6 ( n 1 )

Try it

Try It: The nth Term of Arithmetic Sequences

Find the formula for the n n th term of a sequence where the first term is 7 and the common difference is 9.

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