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Arithmetic Sequences Explicit Formula

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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)

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

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**PRactice -1, -6, -11, -16, -21, ____,____,____**

What are the missing numbers? What is the common difference?

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**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)

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**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 = 𝑎 5 = 2nd term has 1 three 3rd term has 2 threes 4th term has 3 threes 5th term has 4 threes See a pattern?

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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

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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)

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**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

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

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**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 = 𝑛−1 𝑎 𝑛 = 𝑛−150 𝑎 𝑛 =150𝑛

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**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

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