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Arithmetic Sequences

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Arithmetic Series

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Mathematical Induction

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 UNIT 10 : SEQUENCES AND SERIES

 LESSON 3: ARITHMETIC SERIES HOMEWORK QUESTIONS

 

Quick Review

Arithmetic Series:

Recall a sequence such as 2, 3, 8, 13, is called an Arithmetic Sequence. These sequences have the following properties.

        Terms are denoted as t1 , t2 , t3 , referring to term1, term 2, term 3

        The difference between successive terms is constant. ie t2 t1 = t3 t2 = t4 t3 etc

        This difference is called the common difference and denoted using the letter d. Here d = 5.

        The first term is denoted using the letter a. Here a = -2.

        Successive terms are found by adding the common difference, d, to the preceding term. Hence t5 = 13 + 5 = 18 etc.

        The formula for the general term or nth term is tn = a + (n 1)d

        Arithmetic sequences are linear functions with domain the natural numbers N = {1, 2, 3, 4, 5, }

 

Text Box: a = t1 = first term
d = common difference
n = term number
tn = nth term or general term
 

 

 

 

 

 

 


Definition: The sum of the terms of an arithmetic sequence is an Arithmetic Series.

Text Box: n = the number of terms
t1 = a = the first term
tn = l = the last term
d = the common difference
Sn = the sum of the series
Text Box:

 

 

 

 

 

 

 

 

 

Homework Questions:

1. Find the sum of the following arithmetic series.

 

2. In each of the following arithmetic series, determine S60 .

a) 3 + 7 + 11 +

b) -11 + (-2) + 5 + 12 +

c) 2 + 2.5 + 3 +

 

3. For each of the following series, the first and last terms are given. Find the sum in each case.

 

 

5. Find the number of terms in each series below having the given sum.

 

6. For a certain arithmetic series, t5 = 16 and S20 = 650. find a and d.

 

7. Find the first term and common difference for the sequence defined by tn = 5 2n. Find S40.

 

8. On the first day of practice, the track team ran 3 laps of the field. The coach plans to increase the number of laps by two each practice. How many laps will they run on the eleventh practice? How many laps have they run in total after the 11th practice?

 

 

10. The first term of an arithmetic series is 15. The sum of the first 16 terms is 480. Find the common difference and the first 4 terms.

 

11. The first term of an arithmetic series is 4. The sum of the first 15 terms is 1365. Find the common difference and the sum of the first 25 terms.

 

 

 

Solutions:

 

1. Find the sum of the following arithmetic series.

 

Solutions:

a = 1

d = 3

n = ?

tn = 100

Sn = ?

 

 

 

a = 255

d = -4

n = ?

tn = 3

Sn = ?

 

 

 

 

a = -3

d = -4

n = ?

tn = -175

Sn = ?

 

 

 

 

 

2. In each of the following arithmetic series, determine S60 .

a) 3 + 7 + 11 +

b) -11 + (-2) + 5 + 12 +

c) 2 + 2.5 + 3 +

Solutions:

a) 3 + 7 + 11 +

a = 3

d = 4

n = 60

S60 = ?

 

 
 

 

 

 

 

 

 

 


a = -11

d = 7

n= 60

S60 = ?

 
b) -11 + (-2) + 5 + 12 +

 

 

 

 

 

 

 

 

c) 2 + 2.5 + 3 +

a = 2

d = 0.5

n = 60

Sn = ?

 

 

 


 


 


3. For each of the following series, the first and last terms are given. Find the sum in each case.

a = 8

l = ?

n = 11

Sn = ?

 
Solutions:


a = - 4

l = 29

n = 12

Sn =?

 

 

 


a = 11

n = 9

l = -13

Sn =?

 

 

 

a = -10.5

n = 10

l = 3

Sn = ?

 

 

 

 

a = 21

d = 7

n = ?

tn = 147

Sn = ?

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


5. Find the number of terms in each series below having the given sum.

Solutions:

 

 

 

 

 

6. For a certain arithmetic series, t5 = 16 and S20 = 650. find a and d.

Solution:

 

7. Find the first term and common difference for the sequence defined by tn = 5 2n. Find S40.

Solution:

 

8. On the first day of practice, the track team ran 3 laps of the field. The coach plans to increase the number of laps by two each practice. How many laps will they run on the eleventh practice? How many laps have they run in total after the 11th practice?

Solution:

This forms an arithmetic sequence(series) with first term 3 and common difference 2. Find t11.

 

 

 

a = 48

d = - 4

n = 9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


10. The first term of an arithmetic series is 15. The sum of the first 16 terms is 480. Find the common difference and the first 4 terms.

Solution

 


 


11. The first term of an arithmetic series is 4. The sum of the first 15 terms is 1365. Find the common difference and the sum of the first 25 terms.

Solution:

 


 

 


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