Understanding Arithmetic Sequence, Slides of Mathematics

Download Understanding Arithmetic Sequence and more Slides Mathematics in PDF only on Docsity! illustrate arithmetic sequence; determine the nth term of a given arithmetic sequence; find the arithmetic means between term of an arithmetic sequence; determine the sum of the first n terms of a given sequence Objective s Activity 1:REVIEW Find the next 3 terms and the Nth Term of the sequence What is the nth term of the sequence 3, 8,13, 18? 𝒂𝟏=𝟑𝒂𝟐=𝟑+𝟓 𝒂𝟑=𝟑+𝟓+𝟓 𝒂𝟒=𝟑+𝟓+𝟓+𝟓 𝟑 (1) (2) (3) (n-1) 𝒂𝒏=𝟑+𝟓𝒏−𝟓 𝒂𝒏=𝟓𝒏−𝟐 What is the nth term of the sequence -14, -9, -4, 1? 𝒂𝟏=−𝟏𝟒𝒂𝟐=−𝟏𝟒+𝟓 𝒂𝟑=−𝟏𝟒+𝟓+𝟓 𝒂𝟒=−𝟏𝟒+𝟓+𝟓+𝟓 −𝟏𝟒 (1) (2) (3) (n-1) 𝒂𝒏=−𝟏𝟒+𝟓𝒏−𝟓 𝒂𝒏=𝟓𝒏−𝟏𝟗 Arithmetic Sequence The previous activity illustrates sequence where the difference between any two consecutive terms is a constant. This constant is called the common difference and the said sequence is called an arithmetic sequence.4, 7, 10, 13, 16 3 3 3 3 Common difference is 3 An arithmetic sequence is a sequence where every term after the first obtained by adding a constant called the common difference. The sequence 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, respectively. Activity 3: Arith metic or a Not PIP DPDDI99  B. Write the next three terms of the sequence below. 1. 5, 12, 19, 26, … ____ , ____ , ____ 2. 8, 5, 2, -1, … ____ , ____ , ____ 3. -75, -60, -45, -30, … ____ , ____ , ____ 7 -3 33 40 47 -4 -7 -10 -15 0 15 d 15 28 Number of squares 1 2 3 4 5 6 7 8 9 10 Number of matchstic ks                    4 7 1 0 1 3 1 6 1 9 2 2 2 5 2 8 3 1We see that the number of matchsticks forms an arithmetic sequence. Suppose we want to find the 20th, 50th, and 100th terms of the sequence. How do we get them? Do you think a formula would help? If so, we could find a formula for the nth term of the sequence.     ... 4 4+3 4+3+3 4+3+3+ 3 ... In general, the first n terms of an arithmetic sequence with as first term and d as common difference are Consider the table below and complete it. Observe how each term is rewritten.     ... 4+0(3) 4+1(3) 4+2(3) 4+3(3) ... Example 2: 2. Find the of the sequence 4, 7, 10, 13,… Wherein: Solution: 3 57 61 𝑨𝒏=𝑨𝟏+ (𝒏−𝟏 )𝒅Formula: ? 𝑨𝟏=𝟒 20 𝒅=𝟑 Example 3: 3. Find the of the sequence 4, 7, 10, 13,…? Wherein: Solution: 3 147 151 𝑨𝒏=𝑨𝟏+ (𝒏−𝟏 )𝒅Formula: ? 𝑨𝟏=𝟒 50 𝒅=𝟑 4. Which term of the arithmetic sequence 7, 14, 21, 28, … is 224? Formula: 𝑨𝒏=𝑨𝟏+ (𝒏−𝟏 )𝒅 Given: 𝐴1=7 𝑛=? 𝑑=7 𝐴𝑛=224 Solution: 𝐴𝑛=𝐴1+(𝑛−1 )𝑑 224=7+(𝑛−1 )7 224 224=7𝑛 77 𝑛=32 Arithmetic Mean Are the terms between any two nonconsecutive terms of an arithmetic sequence 𝑨𝒏=𝑨𝟏+ (𝒏−𝟏 )𝒅 Formula: wherein: 𝐴1=1𝑠𝑡 𝑡𝑒𝑟𝑚 𝑛=𝑛𝑜.𝑜𝑓 𝑡𝑒𝑟𝑚 𝑑=𝑐𝑜𝑚𝑚𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝐴𝑛= h𝑛𝑡 𝑡𝑒𝑟𝑚 Steps in solving the arithmetic mean 1. Using the formula of arithmetic sequence solve for the value of the common difference 2. To get the value of the missing term add the common difference and the value before the missing term. In arithmetic sequence 2, 4, 6, 8, … the middle terms are called arithmetic mean. 4 is the arithmetic mean between 2 and 6 6 is the arithmetic mean between 4 and 8 4 and 6 are the arithmetic mean between 2 and 8 Example 1: Insert six arithmetic means between 5 and 61. 5, Given: 𝐴1=5𝐴𝑛=61 𝑛=¿ 2 1 4 3 6 5 8 7 61,___,___, ___,___, ___, ___, 𝑑=? Formula: 𝑨𝒏=𝑨𝟏+ (𝒏−𝟏 )𝒅 Solution: 61=5+(8−1 )𝑑 61=5+(7 )𝑑 7𝑑 77¿56 𝒅=𝟖 𝐴2=𝐴1+𝑑 Arithmetic Mean: 𝐴2=5+8 𝐴2=13 𝐴3=𝐴2+𝑑 𝐴3=13+8𝐴3=21 𝐴4=𝐴3+𝑑 𝐴4=29 𝐴4=21+8 𝐴5=𝐴4+𝑑𝐴5=29+8 𝐴5=37 𝐴6=𝐴5+𝑑 𝐴6=37+8 𝐴7=45+8 𝐴5=53 𝐴6=45 𝐴7=𝐴4+𝑑 13 21 29 37 45 53 8 61−5=7𝑑 > Activity 4: What Can You , Insert? EE EEE ES) Insert the indicated number of arithmetic means between the given first and last terms of the following arithmetic sequence. 1. 2 and 14 [3] 2. 9 and 45 [3] 3. 6 and 36 [4] Arithmetic Series Find the sum of the first 20 terms of the arithmetic sequence 15, 19, 23, 27,... Solution 2: Given: 𝐴1=15 𝑑=4 𝑛=20 𝑆𝑛=? Formula: 𝑺𝒏= 𝒏 𝟐 [𝟐𝑨𝟏+(𝒏−𝟏 )𝒅 ] 𝑺𝟐𝟎= 𝟐𝟎 𝟐 [𝟐(𝟏𝟓)+ (𝟐𝟎−𝟏 )𝟒] 𝑺𝟐𝟎=𝟏𝟎 [𝟑𝟎+(𝟏𝟗 )𝟒 ] 𝑺𝟐𝟎=𝟏𝟎 [𝟑𝟎+𝟕𝟔 ] 𝑺𝟐𝟎=𝟏𝟎 [𝟏𝟎𝟔] 𝑺𝟐𝟎=𝟏𝟎𝟔𝟎 Find the sum of the first 24 terms of the arithmetic sequence 15, 10, 5, 0,... Solution 1: Find first the value of by substituting in the formula 𝐴24=15+(24−1 )(−5) 𝐴24=15+(23 )(−5) 𝐴24=15−115 𝐴24=−100 Solving for we substitute in the formula. 𝑆𝑛= 𝑛 2 ( 𝐴1+𝐴𝑛) 𝑆24= 24 2 (15−100 ) 𝑆24=12 (−85 ) 𝑆24=−1020 Find the sum of the first 24 terms of the arithmetic sequence 15, 10, 5, 0,… Solution 2: Given: 𝐴1=15 𝑑=−5 𝑛=24 𝑆𝑛=? Formula: 𝑺𝒏= 𝒏 𝟐 [𝟐𝑨𝟏+(𝒏−𝟏 )𝒅 ] 𝑺𝟐𝟒= 𝟐𝟒 𝟐 [𝟐(𝟏𝟓)+ (𝟐𝟒−𝟏 )(−𝟓)] 𝑺𝟐𝟒=𝟏𝟐 [𝟑𝟎+(𝟐𝟑 )(−𝟓)] 𝑺𝟐𝟒=𝟏𝟐 [𝟑𝟎−𝟏𝟏𝟓 ] 𝑺𝟐𝟒=𝟏𝟐 [−𝟖𝟓 ] 𝑺𝟐𝟒=−𝟏𝟎𝟐𝟎 Check Your Understanding What have you learned from today’s lesson? 1. _____________________ is a sequence where every term after the first term is obtained by adding a constant. 2. The constant number added to the preceding term of the arithmetic sequence is called the ____________________. 3. To determine the common difference between two consecutive terms, we _________________________________ ________________. Arithmetic sequence common difference simply subtract the first term to the second term Check Your Understanding What have you learned from today’s lesson? 4. Arithmetic series is the _______________________________ ___________. 5. ___________________ are the terms between two nonconsecutive terms of an arithmetic sequence. sum of terms in an arit meticsequenc eArithmetic means Evaluation: Solve the following. A. Find the common difference of the given arithmetic sequence. 1. 1, 3, 5, 7, … 2. X, 4x, 7x, ...  B. Find the next three terms of an arithmetic sequence given the following: 1. =5; d=-2 2. =13; d=4  C. Insert 3 arithmetic means between 3 and 23.  D. Find the sum of the arithmetic sequence 3,5,7,9,11