# Sec 2.1.5 How Arithmetic Sequences Work? Generalizing Arithmetic Sequences

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• Sec 2.1.5How Arithmetic Sequences Work?Generalizing Arithmetic Sequences

• Blast from the pastSolve the system of equations:x+9y=33x+21y=-3

• Test 2Thursday Oct 31Happy Halloween!

• So far:Linear function:Constant increase or decrease.Same value is added (or subtracted) to the output as the input increases by one unit.Exponential function:Constant growth or decay by a common ratio.The output is multiplied (or divided) by a common ratio as the input increases by one unit.

• Next few lessonsArithmetic sequences:Constant increase or decrease.Geometric Sequences:Constant growth or decay by a common ratio.

• Some terms you should know before we startDefinition of Counting NumbersThe numbers which are used for counting from one to infinity are called Counting Numbers. More about Counting NumbersCounting numbers are also called as natural numbers. Counting numbers are designated as n.

• Example on Counting Numbers

Identify the counting numbes.A. 30 B. 9.1 C. 0 D. 10E. -2F. 1

• A sequence can be thought of as a function, withthe input numbers consisting of the naturalnumbers, and the output numbers being theterms.

• A sequence in which a constant (d) can be added to each term to get the next term is called an Arithmetic Sequence.The constant (d) is called theCommon Difference.

• To find the common difference (d), subtract any term from one that follows it.2 5 8 11 143333t1t2t3t4t5

• Find the first term and the common difference of each arithmetic sequence.First term (a):4Common difference (d):= 9 4 = 5First term (a):34Common difference (d):-7BE CAREFUL: ALWAYS CHECK TO MAKE SURE THE DIFFERENCE IS THE SAME BETWEEN EACH TERM !

• Now you try!Find the first term and the common difference of each of these arithmetic sequences.b) 11, 23, 35, 47, .a) 1, -4, -9, -14, .d) s-4, 3s-3, 5s-2, 7s-1, ..c) 3x-8, x-8, -x-8, -3x-8

• b) 11, 23, 35, 47, .a) 1, -4, -9, -14, .d) s-4, 3s-3, 5s-2, 7s-1, ..c) 3x-8, x-8, -x-8, -3x-8Answers with solutionsa = 1 andd = a2 - a1 = - 4 - 1 = - 5a = 11 andd = a2 - a1 = 23 - 11 = 12a = 3x-8 andd = a2 - a1 = x 8 (3x 8) = - 2xa = s - 4 andd = a2 - a1 = 3s 3 (s 4) = 2s + 1

• The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence . 3, 7, 11, 15, . We know a = 3 and d = 4t1= a = 3t2= a+d = 3+4 = 7t3= a+d+d = a+2d = 3+2(4) = 11t4 = a+d+d+d = a+3d = 3+3(4) = 15

• The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence . The nth term of an arithmetic sequence is given by:The last # in the sequence/or the # you are looking forFirst termThe position the term is inThe common differencetn = t1 + (n 1) d

• Explicit Formula of a SequenceA formula that allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . .To determine the explicit formula, the pervious term need not be computed.

• Find the 14th term of the arithmetic sequence4, 7, 10, 13,tn = t1 + (n 1) dt14 =The 14th term in this sequence is the number 43!

• Now you try!Find the 10th and 25th term given the following information. Make sure to derive the general formula first and then list ehat you have been provided.b) x+10, x+7, x+4, x+1, .a) 1, 7, 13, 19 .d) The second term is 8 and the common difference is 3c) The first term is 3 and the common difference is -21

• b) x+10, x+7, x+4, x+1,.a) 1, 7, 13, 19 . .d) The second term is 8 and the common difference is 3c) The first term is 3 and the common difference is -21Answers with solutionsa = 1 and d = a2 - a1 = 7 1 = 6tn=a+(n-1)d = 1 + (n-1) 6 = 1+6n-6 So tn = 6n-5t10 = 6(10) 5 = 55t25 = 6(25)-5 = 145a = x+10 and d = a2 - a1 = x+7-(x+10) = -3tn=a+(n-1)d = x+10 + (n-1)(-3) = x+10-3n+3 So tn= x-3n+13t10 = x -3(10)+13 = x - 17t25 = x -3(25)+13 = x - 62a = 3 and d = -21tn=a+(n-1)d = 3 + (n-1) -21 = 3-21n+21 So tn= 24-21nt10 = 24-21(10) = -186 t25 = 24-21(25) = -501a = 8 - 3 = 5 and d = 3tn=a+(n-1)d = 5 + (n-1) 3 = 5+3n-3 So tn = 3n+2t10 = 3(10) +2 = 32 t25 = 3(25)+2 = 77

• Find the 14th term of the arithmetic sequence with first term of 5 and the common difference is 6.tn = a + (n 1) dt14 =The 14th term in this sequence is the number -73!a = 5 and d = -65-6= 5 + (13) * -6= 5 + -78= -73

• In the arithmetic sequence 4,7,10,13,, which term has a value of 301?tn = t1 + (n 1) dThe 100th term in this sequence is 301!

• In the arithmetic sequence 4,7,10,13,, Can a term be 560?tn = t1 + (n 1) d560 is not a term.

• In an arithmetic sequence, term 10 is 33 and term 22 is 3. What are the first four terms of the sequence?The sequence is 60, 57, 54, 51, .HMMM! Two equations you can solve! 33 = a+9d -3 = a+21dBy elimination -36 = 12d -3 = dSOLVE:SOLVE:33 = a + 9d33 = a +9(-3)33 = a 2760 = a

• Recursive FormulaFor a sequences a1, a2, a3, . . . , an, . . . a recursive formula is a formula that requires the computation of all previous terms in order to find the value of an .

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