Arithmetic Sequence

Aims: To know the nth term rule for an arithmetic Sequence.

Be able to find the nth term of an arithmetic Sequence given sufficient information.

Lesson Outcomes

• Name: Know what an arithmetic sequence is.

• Describe: The nth term rule of an arithmetic sequence and what you need to write it down.

• Explain: How to use other information and/or simultaneous equations to find the nth term rule.

Maths

• 2C p257-258 Q1,2,3,6• Next Lesson: Arithmetic Series.

Summing the terms of arithmetic sequences.

• Bhāskara II: 12th Century Indian Mathematician who wrote Lilavati a text containing some of the earliest work on arithmetic (and geometric) sequences.

An “Arithmetic Sequence”...

• ...is a sequence which increases or decreses by the same value (d) between each term, with a starting term (a). E.g. 3,5,7,9,11...

• The terms are...

• Recursive Definition un+1=un+d u1=a

• nth term ur=a+(n-1)d

un u1 u2 u3 u4 u5

Term a a+d a+2d a+3d a+4d

Finding The nth term

• The nth term rule un=a+(n-1)d

• We need to know a and d so lets find the nth term for these arithmetic sequences...

• 3,7,...,15,...,23• 5,...,...,14...• ...,...,4,...,...,...,16• 5th term 9, 100th term 199

Find the nth term rule

• First Term 8 and Common Difference 6

• un = 8+(n-1)6 = 8+6(n-1)

Find the nth term rule

• First Term -9 and Common Difference 7

• un = -9+(n-1)7 = -9 + 7(n-1)

Find the nth term rule

• First Term 5 and Common Difference -8

• un = 5+(n-1)(-8) = 5-8(n-1)

Find the nth term rule

• First Term 0 and Common Difference -9

• un = 0+(n-1)(-9) = 0-(n-1)9 =-9(n-1)

Find the nth term rule

• 7 14 21 28 35 • un = 7 + (n-1)(7) = 7n

Find the nth term rule

• 3 13 23 33 43 • un = 3 + (n-1)(10) = 3 + 10(n-1)

Find the nth term rule

• 8 11 14 17 20 • un = 8 + (n-1)(3) = 8 + 3(n-1)

Find the nth term rule

• 7 2 -3 -8 -13 • un = 7 + (n-1)(-5) = 7 – 5(n-1)

Find the nth term rule

• 4 13 22 31 40 • un = 4 + (n-1)(9) = 4 + 9(n-1)

Find the nth term rule

• u22 = 175 and u28 = 223

• un = 7 + (n-1)(8) = 7 + 8(n-1)

Find the nth term rule

• u1 = 2 and u39 = 268

• un = 2 + (n-1)(7) = 2 + 7(n-1)

Find the nth term rule

• u100 = 801 and u73 = 585

• un = 9 + (n-1)(8) = 9 + 8(n-1)

Find the nth term rule

• u32 = 147 and u35 = 162

• un = -8 + 5(n-1)

Find the nth term rule

• u35 = -70 and u95 = -190

• un = -2 -2(n-1) = -2n

Find the nth term rule

• u45 = -268 and u88 = -526

• un = -4 + (n-1)(-6) = -4 – 6(n-1)

How Many Terms (what is n?)

• How many terms are in the sequence 4, 11, 18, 25, …, 557

• 80

How Many Terms (what is n?)

• How many terms are in the sequence 3, 14, 25, 36, …, 410

• 38

How Many Terms (what is n?)

• How many terms are in the sequence 7, 13, 19, 25, …, 241

• 40

Simultaneous Again!

• An arithmetic sequence has a third term that is three times the first and double the second.

• If the common difference is 8 find the nth term rule.

Trickier Simultaneous

• The first three terms of an arithmetic progression are…

• t, 2t + 2 and 4t – 2 respectively.

• Find the value of t and hence write down the nth term rule.

Special Sequences

• Triangle Numbers

• Factorial Sequence

• Pascal’s Triangle loads of sequences!