Lesson 3.13 Applications of Arithmetic Sequences Concept: Arithmetic Sequences EQ: How do we use arithmetic sequences to solve real world problems? F.LE.2 Vocabulary: Arithmetic sequence, Common difference, Recursive formula, Explicit formula, Null/Zeroth term

Think-Write-Share What do you know about arithmetic sequences? Think back to the lesson over arithmetic sequences and write down everything you remember. Be sure to include formulas.

How to identify an arithmetic sequence In a word problem, look for a common difference being used between each term. Example: Determine whether each situation has a common difference between each term. The height of a plant grows 2 inches each day. The cost of a video game increases by 10% each month. Johnny receives 5 dollars each week for an allowance.

Steps to finding a term for an arithmetic sequence Create a picture of the word problem. Write out the sequence in order to identify the common difference, d, and the first term, 𝒂 𝟏 . Determine which formula would best fit the situation (Recursive or Explicit) REMEMBER: Recursive formula helps us get the next term given the previous term while the explicit formula gives us a specific term. Substitute d and 𝑎 1 in to the formula from step 3. Evaluate the formula for the given term. Interpret the result.

Example 1 You visit the Grand Canyon and drop a penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.  What is the total distance the object will fall in 6 seconds?

Example 1: Picture Sequence Formula Interpretation

Example 1 You visit the Grand Canyon and drop a penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.  What is the total distance the object will fall in 6 seconds? Identify 𝑑 and 𝑎 1 The given sequence is 16, 48, 80, … d=48−16=32

Example 1  

Example 1 Plug 𝑑 and 𝑎 1 in to the formula from step 2. If we plug in 𝑑 and 𝑎 1 from step 1, we get: 𝐴 𝑛 =32 𝑛−1 +16 Simplify: 𝐴 𝑛 =32𝑛−32+16 Distribute 𝐴 𝑛 =32𝑛−16 Combine Like Terms

Example 1 Evaluate the formula for the given value. 𝐴 6 =32(6)−16 𝐴 𝑛 =32𝑛−16 Evaluate the formula for the given value. In the problem, we are looking for the total distance after 6 seconds. Therefore, we will plug in 6 to the equation from step 3. 𝐴 6 =32(6)−16 𝐴 6 =192−16 𝐴 6 =176

𝐴 6 =176 Example 1 Interpret the result The problem referred to the total distance in feet, therefore: After 6 seconds, the penny will have fallen a total distance of 176 feet.

Difference between months Example 2 Tom just bought a new cactus plant for his office. The cactus is currently 3 inches tall and will grow 2 inches every month. How tall will the cactus be after 14 months? Identify 𝑑 and 𝑎 1 𝑑=2 𝑎 1 =5 There is such a thing as a zeroth term or null term. It can be the initial value in certain real world examples. 𝑎 0 =3 After no months have passed, the plant begins at 3 inches tall. Difference between months Height after 1 month

Example 2: Picture Sequence Formula Interpretation

Example 2 𝑑=2 𝑎𝑛𝑑 𝑎 1 =5 Determine which formula would best fit the situation. Since we want the distance after 14 months, we will use the explicit formula which is used to find a specific term. Explicit Formula: 𝐴 𝑛 =𝑑 𝑛−1 + 𝐴 1

Example 2 Plug 𝑑 and 𝑎 1 in to the formula from step 2. If we plug in 𝑑 and 𝑎 1 from step 1, we get: 𝐴 𝑛 =2 𝑛−1 +5 Simplify: 𝐴 𝑛 =2𝑛−2+5 Distribute 𝐴 𝑛 =2𝑛+3 Combine Like Terms

Example 2 Evaluate the formula for the given value. 𝐴 14 =2 14 +3 𝐴 𝑛 =2𝑛+3 Evaluate the formula for the given value. In the problem, we are looking for the height after 14 months. Therefore, we will plug in 14 to the equation from step 3. 𝐴 14 =2 14 +3 𝐴 14 =28+3 𝐴 14 =31

𝐴 14 =31 Example 2 Interpret the result The problem referred to the height in inches, therefore: After 14 months, the cactus will be 31 inches tall.

Difference between weeks Example 3 Kayla starts with $25 in her allowance account. Each week that she does her chores, she receives $10 from her parents. Assuming she doesn’t spend any money, how much money will Kayla have saved after 1 year? Identify 𝑑 and 𝑎 1 𝑑=10 𝑎 1 =35 𝑎 0 =25 Before any weeks have passed, Kayla starts with $25. Difference between weeks Amount after 1 week

Example 3: Picture Sequence Formula Interpretation

Example 3 𝑑=10 𝑎𝑛𝑑 𝑎 1 =35 Determine which formula would best fit the situation. Since we want the distance after 1 year (Which is ___ weeks), we will use the explicit formula which is used to find a specific term. Explicit Formula: 𝐴 𝑛 =𝑑 𝑛−1 + 𝐴 1

Example 3 Plug 𝑑 and 𝑎 1 in to the formula from step 2. If we plug in 𝑑 and 𝑎 1 from step 1, we get: 𝐴 𝑛 =10 𝑛−1 +35 Simplify: 𝐴 𝑛 =10𝑛−10+35 Distribute 𝐴 𝑛 =10𝑛+25 Combine Like Terms

Example 3 Evaluate the formula for the given value. 𝐴 52 =10 52 +25 𝐴 𝑛 =10𝑛+25 Evaluate the formula for the given value. In the problem, we are looking for the amount after 52 weeks. Therefore, we will plug in 52 to the equation from step 3. 𝐴 52 =10 52 +25 𝐴 52 =520+25 𝐴 52 =545

Example 3 Interpret the result 𝐴 52 =545 The problem referred to the amount of money, therefore: After 52 weeks, Kayla will have saved $545.

You Try! A theater has 26 seats in row 1, 29 seats in row 2, and 32 seats in row 3 and so on. If this pattern continues, how many seats are in row 42?

You Try! Picture Sequence Formula Interpretation

Summarizer Write down 3 points that every student should remember in order to solve arithmetic sequences in real world situations.