Fill in the following blanks and submit using the assignment tool in Module 8. You can fill in the blanks and submit as a word document. The second option is to print out the form, write the answers on the printout, scan your answers and submit your work as a PDF file.
I. Sequences
A. Definitions
1. A finite sequence is a function whose domain is the finite set {1, 2, 3, …, n}, where n is a _____________________ number.
2. An ___________________ sequence is a function whose domain is the set of all natural numbers.
3. The range values of a sequence are called the _______________ of the sequence.
B. Factorials
1. The factorial of a positive integer n is denoted as ______. To evaluate n! we multiply all the positive ________________ less than or equal to n.
2. 0! = _________.
3. 4! = _________.
C. Writing the terms of a sequence.
1. If the terms of a sequence are given as a_n=n/(n+1), then
a. a_1=1/(1+1)=1/2
b. a_2=2/(2+1)=2/3
c. a_3=3/(3+1)=3/4
d. a_4=4/5
e. a_5=5/6
f. a6 = ______
2. Write the first four terms of the sequence defined by an = (-1)n(2n – 1)
______, ______, ______, ______
D. Recursive Sequences
1. A recursive sequence is a sequence in which each term is defined using one or more of its ___________________ terms.
2. Write the first five terms of the recursive sequence defined by a1 = 3 and an = 2an-1.
a1 = 3, a2 = 2a1 = 2(3) = 6 a3 = 2a2 = 2(6) = 12 a4 = 2a3 = 2(12) = 24
a5 = _____
3. Write the first five terms of the Fibonacci Sequence: ____, ____, ____, ____, ____.
E. The General term of a sequence.
1. The nth term of a sequence is also known as the _______________ term.
2. To find the formula for the general term, an, we need to find a pattern in relation to n.
a. Consider the terms of the sequence as: an = 5, 10, 15, 20, 25, 30, …
b. List the natural numbers under each term: n = 1, 2. 3, 4, 5, 6, …
c. Next express an in terms of n: in this case an = 5n.
3. Write a formula for the nth term of the infinite sequence 1, 4, 9, 16, 25, 36, …
an = __________ .
II. Series
A. Definitions
1. Let a1, a2, a3, … be a sequence. a1 + a2 + a3 + … + an is called a finite ___________.
2. Then a1 + a2 + a3 + … + an + an-1 + … is called an infinite ______________.
3. The sum of the first n terms of a series is called the nth ________________ sum of the series, denoted by Sn.
B. Given an = 2n – 1, Find S4.
S4 = _______
C. Summation Notation
1. The finite series a1 + a2 + a3 + … + an can be written in summation notation as ∑_(i=1)^n▒a_i .
2. The infinite series a1 + a2 + a3 + … + an + an-1 + … can be written as ∑_(i=1)^∞▒a_i
3. The variable i is called the __________ of summation, the number 1 is called the ____________________ of summation, and n is the ___________________ of summation.
4. The sum ∑_(i=1)^4▒〖〖(i〗^2+1)〗 = 12 + 1 + 22 + 1 + 32 + 1 + 42 + 1 = ________
5. Evaluate: ∑_(i=4)^7▒〖(2i-5)〗 = _________.
D. Finding the sum on a graphing calculator.
1. You should watch the Graphing Calculator Tutorials by David Williamson by going to the URL: http://dtc.pima.edu/~dwilliamson/TI/indexti.html)
a. Watch the topic “Summation” (It is the 2nd to the last topic.)
2. Enter the information in the following format: sum(seq(formula,variable,lower limit, upper limit))
3. Evaluate S25, given an = 3n – 1.
4. The formula is 3n – 1; the variable is n, the lower limit is 1 [note this is the smallest value of n]; and the upper limit is 25 [note this is the largest value of n].
5. So we need to enter the expression: sum(seq(3n-1,n,1,25))
6. Go to the List menu by selecting [2nd][LIST]; then highlight the MATH submenu and select 5: sum(
7. Again go to the List menu; then highlight the OPS submenu and select 5: seq(
8. Enter the information: 3n-1,n,1,25 and close the parenthesis.
9. The sum = ________
10. Note: you can (1) use the variable x in place of n, or you can (2) use the ALPHA button to select N as the variable, or you can (3) change the mode from FUNC to SEQ using the MODE button. This last option automatically changes the variable to n when you press the [X,T,θ,n] key.
III. Arithmetic Sequence
A. A sequence is arithmetic if the __________________ in any two successive terms is constant.
1. In the arithmetic sequence a1, a1+ d, a1 + 2d, a1 + 3d, a1 + 4d, … a1 is the ________ term of the sequence, d is the common __________________, and the general term is an = a1 + (n – 1)d.
2. Given the arithmetic sequence 5, 8, 11, 14, …, find a30.
a. The common difference, d, can be found by subtracting any term and the previous term. For example we can subtract 11 – 8 to get _____ for the common difference, d.
b. If we are looking for a30, then n = 30, and since the first term is 5, a1 = ______.
c. Substituting the values into the formula an = a1 + (n – 1)d, we get a30 = 5 + (30-1)3.
d. Simplifying gives: a30 = _______.
3. Given a4 = -6 and d = -7, find a20.
a. Using the fact that a4 = -6 when n = 4 and d = _____, we substitute into the formula giving us -6 = a1 + (____ – 1)(-7). Solving for a1 we get a1 = 15.
b. Now use the fact that a1 = 15 to solve for a20 in the formula: a20 = ____ +(20-1)(-7)
c. Simplifying gives a20 = -118.
4. Given a5 = 23 and a21 = 87, find a51.
a. Using a5 = 23, we have n = 5. Substituting into the formula gives 23 = a1 + (5-1)d.
b. Likewise, using a21 = 87, we have n = ____. Substituting into the formula gives
_____ = a1 + (21-1)d.
c. Simplifying these two equations gives us the 2×2 system: 23 = a1 + ___ d and
87 = a1 + 20d.
d. Solving this system gives us d = 4 and a1 = 7.
e. Now, to find a51, we use n = _____, a1 = 7 and d = 4. So a51=7 + (51-1)4 = _____.
IV. Arithmetic Series
1. The sum of the first n terms of an arithmetic series is given by the formula S_n=(n(a_1+a_n))/2
2. Find the following sum: ∑_(i=1)^28▒〖(5i+7)〗
a. Since the formula is linear (1st degree polynomial), it is an arithmetic sum.
b. We can also tell it’s arithmetic by expanding a few terms. a1 = 5(1)+7 = 12, a2 = 5(2)+7 = 17, a3 = 5(3)+7 = 22, a4 = 5(4)+7 = 27, … So the sum is 12+17+22+27+… which is arithmetic with a1 = _____ and d = _____.
c. The last term is the 28th term, so n = _____ and a28 = ______.
d. So the sum, S_28=(28(12+147))/2 = _______.
3. Find the following sum: 4 + 5.5 + 7 + 8.5 + 10 + … + 56.5
a. This is an arithmetic sum with a common difference, d = ______.
b. The nth term is 56.5. In order to find out what n is we use the general term formula: 56.5 = _____ + (n-1)(1.5)
c. Now use the nth partial sum formula for an arithmetic series to get S36 = _______.
V. Geometric Sequence
A. Each term of a geometric sequence can be obtained by ________________ the previous term by the same number called the common ____________. We can obtain this number by dividing any term by the previous term.
B. The general term of a geometric sequence is given by an = a1rn-1.
C. Given the sequence 64, 32, 16, 8, …, determine if the sequence is geometric, find the general term, and find a12.
1. determine if the sequence is geometric: a_2/a_1 =32/64=1/2. Note that each term is obtained by multiplying the previous term by _____, therefore the sequence is geometric.
2. find the general term: Since a1 = ____ and r = ____, an = 64(1/2)n-1.
3. find a12: a12 = 64(1/2)11 =2^6 1/2^11 = 1/2^5 = ______.
VI. Geometric Series
A. The nth partial sum of a geometric series is given by: S_n=(a_1 (1-r^n))/(1-r).
B. Find ∑_(i=1)^12▒〖3(2)^(i+1) 〗
1. First expand the summation: 3(2)2 + 3(2)3 + 3(2)4 + … + 3(2)13
2. Compute the first 3 terms: 12 + _____ + _____ + … + 3(2)13
3. Note that each term is obtained by multiplying the previous term by ____.
4. n = the number of terms = _____.
5. r = ____.
6. S_12=(12(1-2^12))/(1-2) = ___________.
C. If ____ < r < _____, the sum of an infinite geometric series is S=a_1/(1-r).
1. If an infinite series has a finite sum, it is said to _______________. If an infinite series does not have a finite sum, it is said to __________________.
2. Determine whether the infinite series -6+4- 8/3+16/9-… converges or diverges. If it converges, find the finite sum.
a. a_2/a_1 =4/(-6)=-2/3. Since r = -2/3, the series ________________ (converges or diverges?)
b. S=(-6)/(1-(-2)/3) = _______.
D. Show that 0.3(14) ̅ is a rational number by expressing it as the ratio of two intergers.
1. To see the pattern write this repeating decimal as 0.3141414141414…
2. Now write it as a sum: 0.3 + 0.014 + 0.00014 + 0.0000014 + 0.000000014 + …
3. Starting with the 2nd term, this is an infinite geometric sum. a1 = 0.014 and r=0.00014/0.014 = _____.
4. Using the formula for an the convergent series we get 0.3(14) ̅ = 0.3 + 0.014/(1-.01) = 0.3 + 0.014/0.99 = 3/10+14/990=297/990+14/990 = ______
VI. Binomial Expansions
A. We call the expansion of (a + b)n a _______________ expansion.
1. In the expansion of (a + b)n there are _________ terms.
2. The sum of the exponents in each term is always equal to ____.
3. The first term is ____ and the last term is ____.
4. As we look at the terms of the expansion from left to right, the exponent of the first variable _______________ by 1, and the exponent of the second variable _________________ by 1.
B. Pascal’s Triangle
1. One way to find the coefficients of a binomial expansion is to use Pascal’s Triangle.
2. The first and last number of each row of Pascal’s Triangle is ____. Every other number is equal to the ____________ of the two numbers directly above it.
3. Give the next row of Pascal’s Triangle
{█(1@1 1@1 2 1@1 3 3 1@1 4 6 4 1@1 5 10 10 5 1@1 6 15 20 15 6 1)} ____, ____, ____, _____, _____, _____, _____,_____
4. Use Pascal’s Triangle to expand (x + 5)5
a. (x + 5)5 = x5 + ___x4∙5 + ___x3∙52 + ___x2∙53 + ___x∙54 + 55
b. = x5 + 25×4 + _____ x3 + 1250×2 + 3125x + 3125
C. Binomial Coefficient Formula
1. The binomial coefficient (█(n@r)), read as n ________ r, is equal to n!/r!(n-r)!
2. The r corresponds to the second exponent of the term. For instance in the 3rd term of the example in B.4. above we have 10×3∙52. Since the second exponent is 2, r = 2 in this term.
3. n is the same as the n in (a + b)n.
4. Evaluate the binomial coefficient (█(6@2)).
a. (█(6@2))=6!/2!(6-2)!=6!/2!4! = ______
5. We can evaluate binomial coefficients using a graphing calculator
a. To evaluate “6 choose 2” ( the previous example) we first type 6 on the home screen.
b. Then select the [MATH] key.
c. Highlight the PRB menu at the far right.
d. Select 3: nCr
e. Then type 2 on the home screen, and press [ENTER].
D. Expand using the Binomial Theorem
1. Using the pattern of exponents discussed earlier and the 1-3 in C above, we can expand a binomial in the form (a + b)n. This technique is given by the Binomial Theorem.
2. Expand (x + 3y)5 using the Binomial Theorem
(x+3y)^5=(█(5@0)) x^5+(█(5@1)) x^4 (3y)^1+(█(5@2)) x^3 (3y)^2+(█(5@3)) x^2 (3y)^3+(█(5@4))x(3y)^4+(█(5@5)) (3y)^5
=1x^5+5x^4 (3y)+10x^3 (9y^2 )+10x^2 (27y^3 )+5x(81y^4 )+1(243y^5 )
= x5 + 15x4y + ____x3y2 + ____x2y3 + _____xy4 + 243y5
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