8.2 Arithmetic Sequences and Partial Sums
A sequence whose consecutive terms have a common difference is called an arithmetic sequence.
a1, a2, a3, a4, ..., an, ...
is arithmetic when there is a number d such that
a2 - a1 = a3 - a2 = a4 - a3 = ⋅⋅⋅ = d.
The number d is the common difference of the sequence.
a1, a2, a3, a4, ..., an, ...
is arithmetic when there is a number d such that
a2 - a1 = a3 - a2 = a4 - a3 = ⋅⋅⋅ = d.
The number d is the common difference of the sequence.
Examples of Arithmetic Sequences
The sequence 1, 4, 9, 16 ..., whose nth term is n², is not arithmetic. The difference between the first two terms is 3, but the difference between the second and third terms is 5.
The nth term of an arithmetic sequence can be derived from the pattern.
a1 = a1
a2 = a1 + d
a3 = a1 + 2d
a4 = a1 + 3d
a5 = a1 + 4d
an = a1 + (n -1)d
The nth term of an arithmetic sequence has the form
an = a1 + (n - 1)d
d is the common difference between consecutive terms of the sequence and a1 is the first term of the sequence.
The nth term of an arithmetic sequence can be derived from the pattern.
a1 = a1
a2 = a1 + d
a3 = a1 + 2d
a4 = a1 + 3d
a5 = a1 + 4d
an = a1 + (n -1)d
The nth term of an arithmetic sequence has the form
an = a1 + (n - 1)d
d is the common difference between consecutive terms of the sequence and a1 is the first term of the sequence.
Finding the nth Term of an Arithmetic Sequence
Writing the Terms of an Arithmetic Sequence
The Sum of a Finite Arithmetic Sequence
There is a simple formula for the sum of a finite arithmetic sequence.
The sum of a finite arithmetic sequence with n terms is given by
Sn = (n/2)(a1 + an)
This formula works only for arithmetic sequences.
There is a simple formula for the sum of a finite arithmetic sequence.
The sum of a finite arithmetic sequence with n terms is given by
Sn = (n/2)(a1 + an)
This formula works only for arithmetic sequences.
Finding the Sum of a Finite Arithmetic Sequence
The sum of the first n terms of an infinite sequence is called the nth partial sum. The nth partial sum of an arithmetic sequence can be found by using the sum of a finite arithmetic sequence.
Finding a Partial Sum of an Arithmetic Sequence
Applications