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# 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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