Algebra 2 Class 5

Sequences and Series

Sienna Choi and Hailey Peck

What are sequences?

A sequence is an ordered list of elements that follow a pattern or rule. We will cover two main types: arithmetic and geometric.

4)

5)

6)

Arithmetic sequences:

3, 7, 11, 15, 19, …

−5, 1, 7, 13, 19, …

−7, 2, 11, 20, …

What is the fifth term in the sequence where the first term is 15 and the common difference is 7?

The tenth term in a sequence is 90, the fourth term is 20, what is the common difference?

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The 4th term of an arithmetic sequence is 17 and the 11th term is 45. What is the 25th term?

1) Find the common difference:

• A sequence of numbers where the distance between each term is constant
• “Common difference”/d
• an = a1+(n-1)d

2) Find the common difference:

3) Write a formula for an:

4)

5)

6)

Arithmetic sequences:

3, 7, 11, 15, 19, …

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4

−5, 1, 7, 13, 19, …

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6

−7, 2, 11, 20, …

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an = 9n - 16

What is the fifth term in the sequence where the first term is 15 and the common difference is 7?

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43

The tenth term in a sequence is 90, the fourth term is 20, what is the common difference?

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35/3

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The 4th term of an arithmetic sequence is 17 and the 11th term is 45. What is the 25th term?

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101

1) Find the common difference:

• A sequence of numbers where the distance between each term is constant
• “Common difference”/d
• an = a1+(n-1)d

2) Find the common difference:

3) Write a formula for an:

2) Find the common ratio

3)Write a formula for an

4)

5)

6)

Geometric sequences:

1) Find the common ratio

2,6,18,54,162,…

3,−6,12,−24,48,…

−2,6,−18,54,…

If the fourth term of a geometric sequence is 27/2, and the first term is 0.5, what is the common ratio?

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If the fourth term of a geometric sequence is 8, and the second term is 1/8, what is the sixth term?

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The 2nd term of a geometric sequence is 12 and the 5th term is 96. Find the 8th term.

• A sequence where each term following the first is found by multiplying the previous term by a constant value
• “Common ratio”/d
• an = a1*r^(n-1)

2) Find the common ratio

3)Write a formula for an:

4)

5)

6)

Geometric sequences:

1) Find the common ratio

2,6,18,54,162,…

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3

3,−6,12,−24,48,…

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-2

−2,6,−18,54,…

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an = -2(-3)^(n-1)

If the fourth term of a geometric sequence is 27/2, and the first term is 0.5, what is the common ratio?

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3

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If the fourth term of a geometric sequence is 8, and the second term is 1/8, what is the sixth term?

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512

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The 2nd term of a geometric sequence is 12 and the 5th term is 96. Find the 8th term.

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768

• A sequence where each term following the first is found by multiplying the previous term by a constant value
• “Common ratio”/d
• an = a1*r^(n-1)

What are series?

A series is the sum of the terms of a sequence. It is often written with sigma notation.

4)

5)

6)

Arithmetic series:

−5, 1, 7, 13, 19

−7, 2, 11, 20

A concert hall has 25 rows of seats. The first row has 20 seats, the second row has 23 seats, and the third row has 26 seats. If this pattern keeps going, how many total seats are in all 25 rows?

Logs are stacked in a pile with 24 logs on the bottom row and 15 on the top row. There are 10 rows in

all with each row having one more log than the one above it. How many logs are in the stack?

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A writer wrote 890 words on the first day, 760 words on the second day and 630 words on the

third day, and so on in an arithmetic sequence. How many words did the writer write in a week?

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1) Find the sum:

• The sum/series of the sequence(numbers where the distance between each term is constant “d”)
• Sn=n(a1+a2)/2 i.e.half of the average the first and last term multiplied by the number of terms

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2) Find the sum:

3) Find the sum

4)

5)

6)

Arithmetic series:

3, 7, 11, 15, 19

−5, 1, 7, 13, 19

−7, 2, 11, 20

A concert hall has 25 rows of seats. The first row has 20 seats, the second row has 23 seats, and the third row has 26 seats. If this pattern keeps going, how many total seats are in all 25 rows?

Logs are stacked in a pile with 24 logs on the bottom row and 15 on the top row. There are 10 rows in

all with each row having one more log than the one above it. How many logs are in the stack?

​

​

A writer wrote 890 words on the first day, 760 words on the second day and 630 words on the

third day, and so on in an arithmetic sequence. How many words did the writer write in a week?

​

1) Find the sum:

• The sum of the arithmetic sequence(numbers where the distance between each term is constant “d”)
• Sn=n(a1+a2)/2 i.e.half of the average the first and last term multiplied by the number of terms

​

2) Find the sum:

3) Find the sum

32

35

26

1400 195 3500

2) Find the sum

3)Find the sum

4)

5)

6)

Geometric series:

1) Find the sum

2,6,18,54,162

3,−6,12,−24,48

−2,6,−18,54

Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second

day, 4 cents on the third day and so on. If the daily wage keeps doubling, what will your total income be

for working 31 days?

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The first year a toy manufacturer introduces a new toy, its sales total $495,000. The company expects

its sales to drop 10% each succeeding year. Find the total expected sales in the first 6 years.

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A company is offering a job with a salary of $30,000 for the first year and a 5% raise each year after

that. If the 5% raise continues every year, find the amount of money you would earn in a 40-year career.

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• The sum of a geometric sequence (where each term following the first is found by multiplying the previous term by a constant value)
• Sn=a1(1-r^n)/(1-r), where r≠1

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• ​

2) Find the sum

3)Find the sum

4)

5)

6)

Geometric series:

1) Find the sum

2,6,18,54,162

3,−6,12,−24,48

−2,6,−18,54

Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second

day, 4 cents on the third day and so on. If the daily wage keeps doubling, what will your total income be

for working 31 days?

​

The first year a toy manufacturer introduces a new toy, its sales total $495,000. The company expects

its sales to drop 10% each succeeding year. Find the total expected sales in the first 6 years.

​

​

A company is offering a job with a salary of $30,000 for the first year and a 5% raise each year after

that. If the 5% raise continues every year, find the amount of money you would earn in a 40-year career.

​

• The sum of a geometric sequence (where each term following the first is found by multiplying the previous term by a constant value)
• Sn=a1(1-r^n)/(1-r), where r≠1

​

• ​

242 33 40

21,474,836.47 2,319,367.05 3,623,993.23

3069