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868 Series and Combinations
Lesson
Arithmetic Series
Chapter 13
BIG IDEA There are several ways to find sums of the successive terms of an arithmetic sequence.
Sums of Consecutive Integers
There is a story the famous mathematician Carl Gauss often told about himself. When he was in third grade, his class misbehaved and the teacher gave the following problem as punishment:
“Add the whole numbers from 1 to 100.”
Gauss solved the problem in almost no time at all. His idea was the following. Let S be the desired sum.
S = 1 + 2 + 3 + … + 98 + 99 + 100
Using the Commutative Property of Addition, the sum can be rewritten in reverse order.
S = 100 + 99 + 98 + ... + 3 + 2 + 1
Now add corresponding terms in the equations above. The sums 1 + 100, 2 + 99, 3 + 98, ... all have the same value!
So 2 S = 101 + 101 + 101 + … + 101 + 101 + 101. 100 terms Thus, 2 S = 100 · 101 and S = 5050.
Gauss wrote only the number 5050 on his slate, having done all the figuring in his head. The teacher (who had hoped the problem would keep the students working for a long time) was quite irritated. However, partly as a result of this incident, the teacher did recognize that Gauss was extraordinary and gave him some advanced books to read. (You read about Gauss’s work in Lesson 11-6 and may recall that he proved the Fundamental Theorem of Algebra at age 18.)
QY
Mental Math
Consider the arithmetic sequence defined by a (^) n = 3 n - 12. a. Find a 1 , a 2 , and a 3. b. Find a 1 + a 2 + a 3. c. Find a 101 , a 102 , and a 103. d. Find a 101 + a 102 + a 103.
Mental Math
Consider the arithmetic sequence defined by a (^) n = 3 n - 12. a. Find a 1 , a 2 , and a 3. b. Find a 1 + a 2 + a 3. c. Find a 101 , a 102 , and a 103. d. Find a 101 + a 102 + a 103.
QY Use Gauss’s method to add the integers from 1 to 40.
QY Use Gauss’s method to add the integers from 1 to 40.
Vocabulary series arithmetic series ∑ , sigma ∑ -notation, sigma notation, summation notation index variable, index
Arithmetic Series 869
Lesson 13-
What Is an Arithmetic Series?
Recall that an arithmetic or linear sequence is a sequence in which the difference between consecutive terms is constant. An arithmetic sequence has the form
a 1 , a 1 + d , a 1 + 2 d , ..., a 1 + ( n - 1) d, ... ,
where a 1 is the first term and d is the constant difference. For example, the odd integers from 1 to 999 form a finite arithmetic sequence with a 1 = 1, n = 500, and d = 2.
A series is an indicated sum of terms of a sequence. For example, for the sequence 1, 2, 3, a series is the indicated sum 1 + 2 + 3. The addends 1, 2, and 3 are the terms of the series. The value, or sum, of the series is 6. In general, the sum of the first n terms of a series a is
S (^) n = a 1 + a 2 + a 3 + ... + a (^) n –2 + a (^) n –1 + a (^) n.
If the terms of a series form an arithmetic sequence, the indicated sum of the terms is called an arithmetic series.
If a is an arithmetic series with first term a 1 and constant difference d , you can find a formula for the value S (^) n of the series by writing the series in two ways:
Start with the first term a 1 and successively add the common difference d.
S (^) n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ... + ( a 1 + ( n - 1) d )
Start with the last term a (^) n and successively subtract the common difference d.
S (^) n = a (^) n + ( a (^) n - d ) + ( a (^) n - 2 d ) + ... + ( a (^) n - (n - 1) d )
Now add corresponding pairs of terms of these two formulas, as Gauss did. Then each of the n pairs has the same sum, a 1 + a (^) n.
S (^) n + S (^) n = ( a 1 + a (^) n ) + ( a 1 + a (^) n ) + ( a 1 + a (^) n ) + … + ( a 1 + a (^) n ) n terms So 2 S (^) n = n ( a 1 + a (^) n ).
Thus, S (^) n = n _ 2 ( a 1 + a (^) n ).
This proves that if a 1 + a 2 + ... + a (^) n is an arithmetic series, then a formula for the value S (^) n of the series is S (^) n = _ n 2 ( a 1 + a (^) n ).
QY
Arithmetic series that involve the sum of consecutive integers from 1 to n lead to a special case of the above formula. In these situations, a 1 = 1 and a (^) n = n , so the sum of the integers from 1 to n
is n _ 2 (1 + n ), or n ___^2 +^ n
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QY Use the formula for Sn to find the sum of the odd integers from 1 to 999.
QY Use the formula for Sn to find the sum of the odd integers from 1 to 999.
Arithmetic Series 871
Lesson 13-
Check Use the formula S (^) n = n_ 2 (a 1 + a (^) n). You need to know how many seats are in the first and last row. In this case, a 1 = 14 and a (^) n = a 20 = 14 + 19 · 2 = 52.
So S 20 =^20 ___ 2 (14 + 52) = ___^202 • 66 = 660.
There are 660 seats in the auditorium. It checks.
Summation Notation
The sum of the first six terms of a sequence a (^) n is
a 1 + a 2 + a 3 + a 4 + a 5 + a 6.
However, when there are many numbers in the series, this notation is too cumbersome. You can shorten this by writing
a 1 + a 2 + …^ + a 6.
It is understood that the terms a 3 , a 4 , and a 5 are included.
This notation can be shortened even further. In a spreadsheet, suppose you have the sum A1 + A2+ A3 + A4 + A5 + A6. That sum can be written as SUM(A1:A6). In algebra, the upper-case Greek letter Σ (sigma) indicates a sum. In Σ -notation, called sigma notation or summation notation, the above sum is written
i = 1
6 ai.
The expression can be read as “the sum of the values of a sub i , for i equals 1 to 6.” The variable i under the ∑ sign is called the index variable, or index. It is common to use the letters i , j , k, or n as index variables. (In summation notation, i is not the complex number √–1.) In this book, index variables have only integer values.
Writing Formulas Using Σ -Notation
The two arithmetic series formulas S (^) n = n _ 2 ( a 1 + a (^) n ) and
S (^) n = n _ 2 (2 a 1 + ( n − 1) d ) can be restated using ∑-notation. Notice that
i is used as the index variable to avoid confusion with the variable n.
Arithmetic Series Formula
In an arithmetic sequence a 1 , a 2 , a 3 , ..., a (^) n with constant difference d,
i = 1
n a (^) i = n_ 2 (a 1 + a (^) n) = _n 2 (2a 1 + (n - 1)d).
READING MATH While you use your index finger to point an object, the index variable is used to point to a value. Thus, a 3 points to the third term of the series named a.
READING MATH While you use your index finger to point an object, the index variable is used to point to a value. Thus, a 3 points to the third term of the series named a.
872 Series and Combinations
Chapter 13
When a (^) i = i , the sequence is the set of all positive integers in increasing order 1, 2, 3, 4, …. Then
i = 1
n i = n _ 2 (1 + n ) = ____ n ( n 2^ + 1).
This is a ∑-notation version of Gauss’s sum.
QY
One advantage of ∑-notation is that you can substitute an expression for a (^) i. For instance, suppose a (^) n = 2 n , the sequence of even positive integers. Then,
i = 1
6
ai = ∑
i = 1
6 (2 i ) = 2 · 1 + 2 · 2 + 2 · 3 + 2 · 4 + 2 · 5 + 2 · 6
= 2 + 4 + 6 + 8 + 10 + 12 = 42.
The sum of the first six positive even integers is 42.
Example 3
Consider ∑
i = 1
500 a (^) i , where a (^) n = 4 n + 6.
a. Write the series without Σ -notation. b. Evaluate the sum. Solution a. Substitute the expression for a (^) i from the explicit formula and use it to write out the terms of the series.
∑ i = 1
500 (4i + 6) = (4 ·?^ + 6) + (4 ·?^ +?^ ) +?^ + ... +? =?^ +?^ +?^ + ... +? b. This is an arithmetic series. The first term is?^. The constant difference is?^. There are?^ terms in the series.
Use the formula ∑
i = 1
n a (^) i = _n 2 (2a 1 + (n - 1)d) to evaluate the series.
∑ i = 1
500
(4i + 6) = __? 2 (2 ·?^ + (?^ - 1)?^ )
=?
QY
Find ∑
i = 1
40 i.
QY
Find ∑
i = 1
40 i.
GUIDEDGUIDED
874 Series and Combinations
Chapter 13
- Multiple Choice 7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 =?^.
A ∑
i = 7
63
i B ∑
i = 7
63
(7 i ) C ∑
i = 1
9 (7 i ) D none of these
12. In ∑
i = 100
300 (5 i ), how many terms are added? ( Be careful! )
In 13 and 14, evaluate the sum.
i =
25
(6 i - 4) 14. ∑
n = –
3 9 · 3 n
APPLYING THE MATHEMATICS
- Finish this sentence: The sum of the first n terms of an arithmetic sequence equals the average of the first and last terms multiplied by?^.
- The Jewish holiday Chanukah is celebrated by lighting candles in a menorah for eight nights. On the first night, two candles are lit, one in the center and one on the right. The two candles are allowed to burn down completely. On the second night, three candles are lit (one in the center, and two others) and are allowed to burn down completely. On each successive night, one more candle is lit than the night before, and all are allowed to burn down completely. How many candles are needed for all eight nights?
- Penny Banks decides to start saving money in a Holiday Club account. At the beginning of January she will deposit $100, and each month thereafter she will increase the deposit amount by $25. How much will Penny deposit during the year?
- a. How many even integers are there from 50 to 100? b. Find the sum of the even integers from 50 to 100.
- a. Write the sum of the squares of the integers from 1 to 100 in ∑-notation. b. Evaluate the sum in Part a.
- a. Translate this statement into an algebraic formula using ∑-notation: The sum of the cubes of the integers from 1 to n is the square of the sum of the integers from 1 to n. b. Verify the statement in Part a when n = 8
- Write the arithmetic mean of the n numbers a 1 , a 2 , a 3 , …, a (^) n using ∑-notation.
Arithmetic Series 875
Lesson 13-
REVIEW
- The function with equation y = k · 2 x^ contains the point (6, 10).
a. What is the value of k? b. Describe the graph of this function. (Lesson 9-1)
- Consider the geometric sequence 16, – 40, 100, – 250, ….
a. Determine the common ratio. b. Write the 5th term. c. Write an explicit formula for the n th term. (Lesson 7-5)
Suppose an account pays 5.75% annual interest compounded monthly. (Lesson 7-4) a. Find the annual percentage yield on the account. b. Find the value of a $1700 deposit after 5 years if no other money is added or withdrawn from the account.
Find an equation for the parabola with a vertical line of symmetry that contains the points (5, 0), (1, 5), and (3, 8). (Lesson 6-6)
a. Find the inverse of 5 2 n 1
b. For what value(s) of n does the inverse not exist? (Lesson 5-5)
EXPLORATION
- The number 9 can be written as an arithmetic series: 9 = 1 + 3 + 5. What other numbers from 1 to 100 can be written as an arithmetic series with three or more positive integer terms?
QY ANSWERS
- S = 1 + 2 + … + 40 S = 40 + 39 + … + 1 2 S = 41 + 41 + … + 41 40 terms 2 S = 40 · 41 S = 820
- S (^) n = __^5002 (1 + 999) = 250 · 1000 = 250,
- 820
- Answers vary. Sample:
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