Arithmetic Sequences Section 4.5

Preparation for Algebra ll 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. Preparation for Algebra ll Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. California Standards

Words to Know Sequence – a list of numbers that often form a pattern Term Term – each number in a sequence arithmetic sequence arithmetic sequence – when the terms of a sequence differ by the same number

Identifying Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. 9, 13, 17, 21, … You add 4 to each term to find the next term. The common difference is 4. 9, 13, 17, 21, … +4 What’s the pattern?

The sequence appears to be an arithmetic sequence with a common difference of 4. If so, the next three terms are 25, 29, 33. 9, 13, 17, 21, … 25, 29, 33, … Identifying Arithmetic Sequences What are the next 3 terms?

Determine whether the sequence appears to be an arithmetic sequence. 10, 8, 5, 1, … The difference between successive terms is not the same. This sequence is not an arithmetic sequence. 10, 8, 5, 1, … –2–3 –4 Identifying Arithmetic Sequences What’s the pattern?

4, 1, – 2, – 5, … Step 1 Find the difference between successive terms. You add –3 to each term to find the next term. The common difference is –3. 4, 1, – 2, – 5, … –3 –3 –3–3 –3–3 Determine whether the sequence appears to be an arithmetic sequence. Identifying Arithmetic Sequences What’s the pattern?

The sequence appears to be an arithmetic sequence with a common difference of – 3. If so, the next three terms are – 8, – 11, – 14. – 8, – 11, – 14, … 4, 1, – 2, – 5, … –3 –3 –3–3 –3–3 What are the next 3 terms? Identifying Arithmetic Sequences

How do we find the 20 th term? … n Position The sequence above starts with 3. The common difference d is 2. You can use the first term, 3, and the common difference, 2, to write a rule for finding a n. 3, 5, 7, 9 … Term a 1 a 2 a 3 a 4 a n

The pattern in the table shows that to find the nth term, add the first term to the product of (n – 1) and the common difference. )( first term + (nth – 1) x difference

Finding the nth Term of an Arithmetic Sequence Find the indicated term of the arithmetic sequence. 16th term? : 4, 8, 12, 16, … 4, 8, 12, 16, … Step 2 Write a rule to find the 16th term. The 16th term is 64. a n = a 1 + (n – 1)d a 16 = 4 + (16 – 1)(4) = 4 + (15)(4) = = 64 ******first term + (nth – 1) x difference

Finding the nth Term of an Arithmetic Sequence Find the indicated term of the arithmetic sequence. The 25th term: a 1 = –5; d = –2 The 25th term is –53. a n = a 1 + (n – 1)d a 25 = – 5 + (25 – 1)( – 2) = – 5 + (24)( – 2) = – 5 + ( – 48) = – 53 -5, -7, -9, -11, -13, … ******first term + (nth – 1) x difference

Find the indicated term of the arithmetic sequence. 60th term: 11, 5, –1, –7, … 11, 5, –1, –7, … –6 –6 –6 Step 2 Write a rule to find the 60th term. The 60th term is –343. a n = a 1 + (n – 1)d a 60 = 11 + (60 – 1)( – 6) = 11 + (59)( – 6) = 11 + ( – 354) = – 343 Finding the nth Term of an Arithmetic Sequence ******first term + (nth – 1) x difference

Lesson Quiz: Part I Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence. 1. 3, 9, 27, 81, … not arithmetic 2. 5, 6.5, 8, 9.5, … arithmetic; 1.5; 11, 12.5, 14

Lesson Quiz: Part II Find the indicated term of each arithmetic sequence rd term: –4, –7, –10, –13, … 4. 40th term: 2, 7, 12, 17, … 5. 7th term: a 1 = – 12, d = th term: a 1 = 3.2, d = 2.6 –