Arithmetic Sequences Class 10 Notes | EduRev

Created by: Paras Saxena

Class 10 : Arithmetic Sequences Class 10 Notes | EduRev

 Page 1


ARITHMETIC 
SEQUENCES 
These are sequences where the difference 
between successive terms of a sequence 
is always the same number.  This number 
is called the common difference. 
Page 2


ARITHMETIC 
SEQUENCES 
These are sequences where the difference 
between successive terms of a sequence 
is always the same number.  This number 
is called the common difference. 
3, 7, 11, 15, 19 … 
Notice in this sequence that if we find the difference 
between any term and the term before it we always get 4.  
4 is then called the common difference and is denoted 
with the letter d.   
d = 4 
To get to the next term in the sequence we would add 4 
so a recursive formula for this sequence is:   
4
1
? ?
? n n
a a
The first term in the sequence would be a
1
 which is 
sometimes just written as a. 
a = 3 
Page 3


ARITHMETIC 
SEQUENCES 
These are sequences where the difference 
between successive terms of a sequence 
is always the same number.  This number 
is called the common difference. 
3, 7, 11, 15, 19 … 
Notice in this sequence that if we find the difference 
between any term and the term before it we always get 4.  
4 is then called the common difference and is denoted 
with the letter d.   
d = 4 
To get to the next term in the sequence we would add 4 
so a recursive formula for this sequence is:   
4
1
? ?
? n n
a a
The first term in the sequence would be a
1
 which is 
sometimes just written as a. 
a = 3 
3, 7, 11, 15, 19 … 
+4 +4 +4 +4 
Each time you want another term in the sequence you’d 
add d.  This would mean the second term was the first 
term plus d.  The third term is the first term plus d plus d 
(added twice).  The fourth term is the first term plus d plus 
d plus d (added three times).  So you can see to get the 
nth term we’d take the first term and add d (n - 1) times. 
d = 4 
? ?d n a a
n
1 ? ? ?
Try this to get the 5th term.   
a = 3 
? ? 19 16 3 4 1 5 3
5
? ? ? ? ? ? a
Page 4


ARITHMETIC 
SEQUENCES 
These are sequences where the difference 
between successive terms of a sequence 
is always the same number.  This number 
is called the common difference. 
3, 7, 11, 15, 19 … 
Notice in this sequence that if we find the difference 
between any term and the term before it we always get 4.  
4 is then called the common difference and is denoted 
with the letter d.   
d = 4 
To get to the next term in the sequence we would add 4 
so a recursive formula for this sequence is:   
4
1
? ?
? n n
a a
The first term in the sequence would be a
1
 which is 
sometimes just written as a. 
a = 3 
3, 7, 11, 15, 19 … 
+4 +4 +4 +4 
Each time you want another term in the sequence you’d 
add d.  This would mean the second term was the first 
term plus d.  The third term is the first term plus d plus d 
(added twice).  The fourth term is the first term plus d plus 
d plus d (added three times).  So you can see to get the 
nth term we’d take the first term and add d (n - 1) times. 
d = 4 
? ?d n a a
n
1 ? ? ?
Try this to get the 5th term.   
a = 3 
? ? 19 16 3 4 1 5 3
5
? ? ? ? ? ? a
Let’s look at a formula for an arithmetic sequence and see 
what it tells us. 
? ? 1 4 ? n
Subbing in the set of positive integers we get: 
3, 7, 11, 15, 19, … 
What is the 
common 
difference? 
d = 4 
you can see what the 
common difference 
will be in the formula 
We can think of this as 
a “compensating term”.  
Without it the sequence 
would start at 4 but this 
gets it started where we 
want it. 
4n would generate the multiples of 4.  
With the - 1 on the end, everything is back 
one.  What would you do if you wanted 
the sequence 2, 6, 10, 14, 18, . . .? 
? ? 2 4 ? n
Page 5


ARITHMETIC 
SEQUENCES 
These are sequences where the difference 
between successive terms of a sequence 
is always the same number.  This number 
is called the common difference. 
3, 7, 11, 15, 19 … 
Notice in this sequence that if we find the difference 
between any term and the term before it we always get 4.  
4 is then called the common difference and is denoted 
with the letter d.   
d = 4 
To get to the next term in the sequence we would add 4 
so a recursive formula for this sequence is:   
4
1
? ?
? n n
a a
The first term in the sequence would be a
1
 which is 
sometimes just written as a. 
a = 3 
3, 7, 11, 15, 19 … 
+4 +4 +4 +4 
Each time you want another term in the sequence you’d 
add d.  This would mean the second term was the first 
term plus d.  The third term is the first term plus d plus d 
(added twice).  The fourth term is the first term plus d plus 
d plus d (added three times).  So you can see to get the 
nth term we’d take the first term and add d (n - 1) times. 
d = 4 
? ?d n a a
n
1 ? ? ?
Try this to get the 5th term.   
a = 3 
? ? 19 16 3 4 1 5 3
5
? ? ? ? ? ? a
Let’s look at a formula for an arithmetic sequence and see 
what it tells us. 
? ? 1 4 ? n
Subbing in the set of positive integers we get: 
3, 7, 11, 15, 19, … 
What is the 
common 
difference? 
d = 4 
you can see what the 
common difference 
will be in the formula 
We can think of this as 
a “compensating term”.  
Without it the sequence 
would start at 4 but this 
gets it started where we 
want it. 
4n would generate the multiples of 4.  
With the - 1 on the end, everything is back 
one.  What would you do if you wanted 
the sequence 2, 6, 10, 14, 18, . . .? 
? ? 2 4 ? n
Find the nth term of the arithmetic sequence when a = 6 and d = -2 
If we use -2n we will generate a sequence whose 
common difference is -2, but this sequence starts at -2 
(put 1 in for n to get first term to see this).  We want 
ours to start at 6.  We then need the “compensating 
term”.  If we are at -2 but want 6, we’d need to add 8. 
? ? 8 2 ? ? n
Check it out by putting in the first few positive 
integers and verifying that it generates our sequence. 
6, 4, 2, 0, -2, . . . 
Sure enough---it starts at 6 and 
has a common difference of -2 
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