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Arithmetic sequence

This document defines and provides examples of arithmetic sequences. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant value, called the common difference, to the preceding term. The document provides the arithmetic sequence formula and examples of using it to find terms and sums of arithmetic sequences. Various activities are presented for students to practice identifying, describing, and working with arithmetic sequences.

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A sequence is a function whose domain is the finite set {1,23, …, n} or the infinite set {1,2 ,3,…}.
Example: n 1 2 3 4 5 a(n) 3 4 5 6 7
Example: n 1 2 3 4 … a(n) 2 4 6 8 …
Activity: Term after TermFind the first five terms of the sequence given the nth term.
1. π‘Ž 𝑛= 𝑛 + 4 2. π‘Ž 𝑛= 2𝑛 βˆ’ 1 3. π‘Ž 𝑛= 12 βˆ’ 3𝑛 4. π‘Ž 𝑛= 3 𝑛 5. π‘Ž 𝑛= (βˆ’2) 𝑛
Activity: Getting to Know You What is the nth term for each sequence below?
1. 3, 5, 7, 9, 11, … 2. 3, 4, 5, 6, 7, … 3. 2, 4, 8, 16, 32, … 4. -1, 1, -1, 1, -1, … 5. 1, Β½ , 1/3, ΒΌ, 1/5,…
ARITHMETIC SEQUENCE
An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called the common difference.
The sequences 1,4,7,10,... and 15, 11, 7, 3,... are examples of arithmetic sequences since each one has a common difference of 3 and -4.
Arithmetic Rule an= a1+(n - 1)d β€’a1 is the first term in the sequence β€’n is the number of the term you are trying to determine β€’d is the common difference β€’an is the value of the term that are looking for
Try this… Use the arithmetic formula to determine the 100th term of the following sequence: 75, 25, -25, -75, -125,…
an = a1 + (n - 1)d = 75 + (100 – 1)(-50) = -4875a1 = 75 n = 100 d = -50
Try this… Use the arithmetic formula to determine the 10th term of the following sequence: 5, 12, 19, 26,…
an = a1 + (n - 1)d = 5 + (10 – 1)(7) = 68a1 = 5 n = 10 d = 7
What is missing? 1. 3, 12, 21, __ , __ , __30 39 48 2. 8, 3, -2, __ , __-7 -12 3. 5, 12, __ , 26, __19 33
What is missing? 4. __, __,__, 8 , 12 , 16-4 0 4 5. -1, __ , __,__, 31, 397 15 6. 13,__ ,__,__,-11.-177 1 23 -5
Insert three terms between 2 and 34 of an arithmetic sequence.
ARITHMETIC MEANS The terms between any two consecutive terms of an arithmetic sequence.
Insert 4 arithmetic means between 5 and 25.
Solution: Since we are required to insert 4 terms, then there will be 6 terms in all.
Let π‘Ž1 = 5 and π‘Ž6 = 25 5, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, 25
Let π‘Ž6 = π‘Ž1 + 6 βˆ’ 1 𝑑 5, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, 25 25 = 5 + 5 𝑑 20 βˆ’ 5 = 5𝑑 4 = 𝑑
5, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, 25 π‘Ž2 = π‘Ž1 + (1)(𝑑) n = 2 (2 – 1) = 1 π‘Ž2 = 5 + 1 4 = 9 π‘Ž3 = 5 + (2)(4)=13 π‘Ž4 = 5 + 3 4 = 17 π‘Ž5 = 5 + 4 4 = 21
Insert 2 arithmetic means between Β½ and 2.
ARITHMETIC SUM Sum of the terms in an arithmetic sequence.
What is 1 + 2 + 3 + …+ 50 + 51 +… + 98 + 99 +100?
Formula: 𝑆 𝑛 = 𝑛 2 (π‘Ž1 + π‘Ž 𝑛) 𝑆 𝑛 = 𝑛 2 [2π‘Ž1 + 𝑛 βˆ’ 1 𝑑] π‘Ž 𝑛 = π‘Ž1 + 𝑛 βˆ’ 1 𝑑
Example: Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17,…
Using the formula: 𝑆 𝑛 = 𝑛 2 [2π‘Ž1 + 𝑛 βˆ’ 1 𝑑] 𝑆10 = 10 2 [2(5) + 10 βˆ’ 1 4] d = 9 - 5 𝑆10 = 5 10 + 9 4 = 5 46 = πŸπŸ‘πŸŽ
Example: Find the sum of the first 20 terms of the arithmetic sequence -2, -5, -8, -11, … - 610
Activity: How well do you know me?
Which of the following sequences is an arithmetic sequence? Why?
1. 3, 7, 11, 15, 19 2. 4, 16, 64, 256 3. 48, 24, 12, 6, 3, … 4. 1, 4, 9, 16, 25, 36 5. 1, Β½ , 0, - Β½
6. – 2, 4, - 8, 16, … 7. 1, 0, -1, -2, -3 8. Β½ , 1/3, ΒΌ , 1/5 9. 3x, x, x/3, x/9 10. 9.5, 7.5, 5.5, 3.5
Find the 25th term of the arithmetic sequence 3, 7, 11, 15, 19,…
The second term of the arithmetic sequence is 24 and the fifth term is 3. Find the first term and the common difference
Give the arithmetic sequence of 5 terms if the first term is 8 and the last term is 100.
What must be the value of k so that 5k – 3, k +2, and 3k – 11 will form an arithmetic sequence?