The numbers in sequences are called terms.You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

The domain gives the relative position of each term.1 2 3 4 5 DOMAIN:3 6 9 12 15RANGE:The range gives the terms of the sequence.This is a finite sequence having the rulean = 3n,where an represents the nth term of the sequence.

Write the first six terms of the sequence an = 2n + 3.SOLUTIONa 1 = 2(1) + 3 = 51st term2nd term3rd term4th term6th terma 2 = 2(2) + 3 = 7a 3 = 2(3) + 3 = 9a 4 = 2(4) + 3 = 11a 5 = 2(5) + 3 = 13a 6 = 2(6) + 3 = 155th term

Write the first six terms of the sequence f (n) = (2) n 1 .SOLUTIONf (1) = (2) 1 1 = 11st term2nd term3rd term4th term6th termf (2) = (2) 2 1 = 2f (3) = (2) 3 1 = 4f (4) = (2) 4 1 = 8f (5) = (2) 5 1 = 16f (6) = (2) 6 1 = 325th term

If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence.Describe the pattern, write the next term, and write a rule for the n th term of the sequence

SOLUTION1 2 3 4

2 6 12 20SOLUTIONA rule for the nth term is f (n) = n (n+1).5(5 +1)Describe the pattern, write the next term, and write a rule for the n th term of the sequence.2, 6, 12 , 20,.1 2 3 4

You can graph a sequence by letting the horizontal axisrepresent the position numbers (the domain) and the vertical axis represent the terms (the range).

You work in the produce department of a grocery store and are stacking oranges in the shape of square pyramid with ten layers. Write a rule for the number of oranges in each layer. Graph the sequence.

SOLUTIONThe diagram below shows the first three layers of the stack. Let an represent the number of oranges in layer n.From the diagram, you can see that an = n 2

Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).

USING SERIES. . .You can use summation notation to write a series. For example, for the finite series shown above, you can writeWhen the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite.

USING SERIESIs read as the sum from i equals 1 to 5 of 3i.index of summationlower limit of summationupper limit of summation

USING SERIESSummation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written .Summation notation for an infinite series is similar to that for a finite series. For example, for the infinite series shown earlier, you can write:The infinity symbol, , indicates that the series continues without end.

USING SERIESThe index of summation does not have to be i. Any letter can be used. Also, the index does not have to begin at 1.

Write the series with summation notation.SOLUTIONNotice that the first term is 5 (1), the second is 5 (2), the third is 5 (3), and the last is 5 (20). So the terms of the series can be written as:ai = 5i where i = 1, 2, 3, . . . , 20

Notice that for each term the denominator of the fractionis 1 more than the numerator. So, the terms of the series can be written as:Write the series with summation notation.SOLUTION

The sum of the terms of a finite sequence can be found by simply adding the terms. For sequences with many terms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.

gives the sum of positive integers from 1 to n .gives the sum of squares of positive integers from 1 to n.gives the sum of n 1s .

RETAIL DISPLAYS How many oranges are in a square pyramid 10 layers high?

SOLUTIONYou know from the earlier example that the i th term of the series is given by ai = i 2, where i = 1, 2, 3, . . . , 10.= 385There are 385 oranges in the stack.