29.2 Arithmetic SequencesA sequence is arithmetic if the differences betweenconsecutive terms are the same.That common difference is called d.For Ex. 7, 11, 15, 19, … , 4n or2, -3, -8, -13, … , 7 - 5n are arith. seq.’sLet’s take a look at the first example. What is d, thecommon difference? What is a1?a1 = 7a4 = 7 + 3(4) = 19a2 = = 11Do you see a pattern?What is a5?What is an?a3 = 7 + 2(4) = 15What is the 35th term?
3Can you come up with an equation that will give you the nth term of any arithmetic sequence?an = a1 + (n - 1)dFind a formula for the nth term of the arithmetic sequencewhose common difference is 5 and whose 2nd term is 12.What is the 18th term of the sequence?What do we know?That d = 5 and a2 = 12a2 = a1 + (2-1)d or a1 + dSubstitute in for a2 and dand solve for a112 = a and a1 = 7How now, do we find the 18th term?a18 = a1 + 17d = (5) = 92
4Find the ninth term of the arithmetic sequence whose first two terms are 2 and 9.What is the common difference, d?d = 7Now we can find the 9th term because we know a1 and d.Find a9.a9 = 2 + 8(7) = 58
5The fourth term of an arithmetic sequence is 20, and the 13th is Write the first 4 terms of this sequence.Write the equations of any 4th and 13th terms of anyarithmetic sequence.a4 = a1 + 3da13 = a1 + 12dNow fill in what we know anduse ellimination to find a1 and d.So the first four terms are:5, 10, 15, 20Homework 1-37 odd
6The sum of a finite arithmetic sequence Ex. Add the numbers from 1 to 100.
7Find the sum.It would take way too longto do this by hand. Usingthe formula just given, wecan do it in seconds.First, we have to find a1 and a150.a1 =a150 =Now substitute thoseinto the formula.
8Insert 3 arithmetic means between 4 and 15. 4 ____ ____ ____ 15We need to find the common difference.Write the equation for a5.a5 = a1+ 4d15 = 4 + 4dSo the sequence is?
9An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 in the thesecond row, 22 in the third row, and so on.How many seats are there in all 20 rows?So, what do we know? Do we know a1? d?Do we know how many seats are in the lastrow? a20a20 = (1) = 39 seats in the last row.Now find S.= 590Homework odd