Arithmetic Sequences Explicit Formula

Arithmetic Sequence A sequence of numbers where each number is increased or decreased by the same amount to get the next term The “same amount” is often known as the common difference (d)

Practice 2, 5, 8, 11, 14, ____,____,_____ What are the missing numbers? What is the common difference?

PRactice -1, -6, -11, -16, -21, ____,____,____ What are the missing numbers? What is the common difference?

What if I want the 50th Term? 2, 5, 8, 11, 14, … What if I want 𝑎 50 ? Do I really want to add three that many times until I find the 50th term? There is a formula for finding the nth term. (finding any term we want)

2, 5, 8, 11, 14 Lets see if we can figure out the formula on our own. 𝑎 1 =2 𝑎 2 =2+3 𝑎 3 =2+3+3 𝑎 4 =2+3+3+3 𝑎 5 =2+3+3+3+3 2nd term has 1 three 3rd term has 2 threes 4th term has 3 threes 5th term has 4 threes See a pattern?

The formula In the formula, 3 is added one less times than the number we are looking for (since if wasn’t part of the first term) How many times would I add 3 if I wanted to find 𝑎 50 ? 49 How many times would I add 3 if I wanted to find 𝑎 132 ? 131

The formula So to find 𝑎 50 , I would take d, which is 3 and add it to my 𝑎 1 , which is 2, 49 times. That is a lot of adding. But if we remember from elementary school that repetitive adding is the same as multiplying, our job will be easier. 𝑎 50 =2+3(49)

There are four parts to this equation. The formula The 50th term formula is 𝑎 50 =2+3(49) , but what about for any term? 𝑎 𝑛 = 𝑎 1 +𝑑(𝑛−1) There are four parts to this equation. Multiply by one less than whatever term you want Common Difference Any term 1st term

The Explicit Formula This is sometimes referred to as the EXPLICIT formula. 𝑎 𝑛 = 𝑎 1 +𝑑(𝑛−1)

Write the explicit formula for each arithmetic sequence: 𝑎 𝑛 = 𝑎 1 +𝑑(𝑛−1) Write the explicit formula for each arithmetic sequence: 150, 300, 450, 600, 750, … 𝑎 𝑛 =150+150 𝑛−1 𝑎 𝑛 =150+150𝑛−150 𝑎 𝑛 =150𝑛

Write the explicit formula for each arithmetic sequence: 𝑎 𝑛 = 𝑎 1 +𝑑(𝑛−1) Write the explicit formula for each arithmetic sequence: 17, 10, 3, -4, -11, -18, … 𝑎 𝑛 =17−7 𝑛−1 𝑎 𝑛 =17−7𝑛+7 𝑎 𝑛 =24−7𝑛