Finding the Common DifferenceDetermine the common difference dfor the arithmetic sequence.
Since the sequence is arithmetic, dis the difference between any two adjacent
terms: We arbitrarily choose the terms 10 and 17.
or 7
Verify that anytwo adjacent terms would give the same result. NOW TRY
Writing the Terms of a Sequence from the First Term
and the Common DifferenceWrite the first five terms of the arithmetic sequence with first term 3 and common
difference
The second term is found by adding to the first term 3, getting 1. For the next
term, add to 1, and so on. The first five terms are
NOW TRYOBJECTIVE 2 Find the general term of an arithmetic sequence.Generalizing
fromExample 2,if we know the first term and the common difference dof an
arithmetic sequence, then the sequence is completely defined as
.
Writing the terms of the sequence in this way suggests the following formula for .an
a 1 , a 2 =a 1 + d, a 3 =a 1 + 2 d, a 4 =a 1 + 3 d,Á
a 1
3, 1, -1, -3, -5.
- 2
- 2
- 2.
EXAMPLE 2
d= 17 - 10,
an+ 1 - an.
- 11, -4, 3, 10, 17, 24,Á
EXAMPLE 1
SECTION 12.2 Arithmetic Sequences 685
General Term of an Arithmetic SequenceThe general term of an arithmetic sequence with first term and common dif-
ference dis
ana 1 1 n 12 d.
a 1
Since is a linear function in n, any linear ex-
pression of the form where kand care real numbers, defines an arithmetic
sequence.
Finding the General Term of an Arithmetic SequenceDetermine the general term of the arithmetic sequence.
Then use the general term to find
The first term is
,or 3. Let
Now find
Formula for
Let
Distributive propertyan= 3 n- 12 Combine like terms.
an=- 9 + 3 n- 3
an= - 9 + 1 n- 12132 a 1 =-9, d=3.
a (^) n=a 1 + 1 n- 12 d an
an.
d= - 3 - 1 - 62 d=a 3 - a 2.
a 1 = - 9.
a 20.
- 9, -6, -3, 0, 3, 6,Á
EXAMPLE 3
kn+ c,
an=a 1 + 1 n- 12 d=dn+ 1 a 1 - d 2
NOW TRY
EXERCISE 1
Determine the common dif-
ference dfor the arithmetic
sequence.
- 4, -13, -22, -31, -40,Á
NOW TRY ANSWERS
- d=- 9
NOW TRY
EXERCISE 2
Write the first five terms of
the arithmetic sequence with
first term 10 and common
difference .- 8
- 10, 2, -6, -14, - 22