Transcript 7.2
7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for an arithmetic sequence? How do you find the rule when given two terms? Arithmetic Sequence: • The difference between consecutive terms is constant (or the same). • The constant difference is also known as the common difference (d). Find the common difference by subtracting the term on the left from the next term on the right. Example: Decide whether each sequence is arithmetic. • • • • • • • -10,-6,-2,0,2,6,10,… -6--10=4 -2--6=4 0--2=2 2-0=2 6-2=4 10-6=4 Not arithmetic (because the differences are not the same) • • • • • 5,11,17,23,29,… 11-5=6 17-11=6 23-17=6 29-23=6 • Arithmetic (common difference is 6) Rule for an Arithmetic Sequence Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12. • There is a common difference where d=15, therefore the sequence is arithmetic. • Use an=a1+(n-1)d an=32+(n-1)(15) an=32+15n-15 an=17+15n a12=17+15(12)=197 a. One term of an arithmetic sequence is a19 = 48. The common difference is d = 3. a. Write a rule for the nth term. SOLUTION a. Use the general rule to find the first term. an = a1 + (n – 1) Write general rule. d a19 = a1 + (19 – 1) d Substitute 19 for n Substitute 48 for a19 and 3 for d. 48 = a1 + 18(3) Solve for a1. – 6 = a1 So, a rule for the nth term is: an = a1 + (n – 1) d = – 6 + (n – 1) 3 Write general rule. Substitute – 6 for a1 and 3 for d. Simplify. b. Graph the sequence. One term of an arithmetic sequence is a19 = 48. The common difference is d =3. b. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. Example: One term of an arithmetic sequence is a8=50. The common difference is 0.25. Write a rule for the nth term. • Use an=a1+(n-1)d to find the 1st term! a8=a1+(8-1)(.25) 50=a1+(7)(.25) 50=a1+1.75 48.25=a1 * Now, use an=a1+(n-1)d to find the rule. an=48.25+(n-1)(.25) an=48.25+.25n-.25 an=48+.25n Now graph an=48+.25n. • Just like yesterday, remember to graph the ordered pairs of the form (n,an) • So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc. Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term. • Begin by writing 2 equations; one for each term given. a5=a1+(5-1)d OR 10=a1+4d And a30=a1+(30-1)d OR 110=a1+29d • Now use the 2 equations to solve for a1 & d. 10=a1+4d 110=a1+29d (subtract the equations to cancel a1) -100= -25d So, d=4 and a1=-6 (now find the rule) an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n Example (part 2): using the rule an=-10+4n, write the value of n for which an=-2. -2=-10+4n 8=4n 2=n Two terms of an arithmetic sequence are a8 = 21 and a27 = 97. Find a rule for the nth term. SOLUTION STEP 1 Write a system of equations using an = a1 + (n – 1)d and substituting 27 for n (Eq 1) and then 8 for n (Eq 2). a27 = a1 + (27 – 1)d Equation 1 97 = a1 + 26d Equation 2 21 = a1 + 7d a8 = a1 + (8 – 1)d Subtract. STEP 2 Solve the system. 76 = 19d Solve for d. 4=d Substitute for d in Eq 1. 97 = a1 + 26(4) 27 = a1 Solve for a1. STEP 3 Find a rule for an. an = a1 + (n – 1)d Write general rule. Substitute for = – 7 + (n – 1)4 a1 and d. = – 11 + 4n Simplify. • What is an arithmetic sequence? The difference between consecutive terms is a constant • What is the rule for an arithmetic sequence? an=a1+(n-1)d • How do you find the rule when given two terms? Write two equations with two unknowns and use linear combination to solve for the variables. 7.2 Assignment p. 446, 3-35 odd Analyze Arithmetic Sequences and Series day 2 What is the formula for find the sum of a finite arithmetic series? Arithmetic Series • The sum of the terms in an arithmetic sequence 1st Term • The formula to find the sum of a finite arithmetic series is: Last Term a1 an S n n 2 # of terms Example: Consider the arithmetic series 20+18+16+14+… . • Find the sum of the 1st • Find n such that Sn=-760 25 terms. a1 an S n n • First find the rule for 2 the nth term. • an=22-2n 20 (22 2n) 760 n 2 • So, a25 = -28 (last term) a1 an S n n 2 20 28 S 25 25 S 25 25(4) 100 2 20 (22 2n) 760 n 2 -1520=n(20+22-2n) -1520=-2n2+42n 2n2-42n-1520=0 n2-21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive solution! SOLUTION a1 = 3 + 5(1) = 8 a20 = 3 + 5(20) =103 ( 8 +) 103 S20 = 20 2 = 1110 Identify first term. Identify last term. Write rule for S20, substituting 8 for a1 and 103 for a20. Simplify. ANSWER The correct answer is C. House Of Cards You are making a house of cards similar to the one shown a. Write a rule for the number of cards in the nth row if the top row is row 1. SOLUTION a. Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12, . . . . These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is: an = a1 + (n– 1) = d Write general rule. Substitute 3 for a1 and 3 for d. = 3 + (n – 1)3 Simplify. = 3n House Of Cards You are making a house of cards similar to the one shown b. What is the total number of cards if the house of cards has 14 rows? SOLUTION b. Find the sum of an arithmetic series with first term a1 = 3 and last term a14 = 3(14) = 42. Total number of cards = S14 5. Find the sum of the arithmetic series (2 + 7i). i=1 SOLUTION a1 = 2 + 7(1) = 9 a12 = 2 + (7)(12) = 2 + 84 = 86 ( a1 )+ an Sn = n 2 S12 = 570 ANSWER S12 = 570 What is the formula for find the sum of a finite arithmetic series? a1 an S n n 2 7.2 Assignment: p. 446 40-48 all, 63-64