9.2 Arithmetic Sequences And Series

Arithmetic Sequences

An arithmetic sequenceA sequence of numbers where each successive number is the sum of the previous number and some constant d., or arithmetic progressionUsed when referring to an arithmetic sequence., is a sequence of numbers where each successive number is the sum of the previous number and some constant d.

an=an−1+d   Arithmetic Sequence

And because an−an−1=d, the constant d is called the common differenceThe constant d that is obtained from subtracting any two successive terms of an arithmetic sequence; an−an−1=d.. For example, the sequence of positive odd integers is an arithmetic sequence,

1,3,5,7,9,…

Here a1=1 and the difference between any two successive terms is 2. We can construct the general term an=an−1+2 where,

a1=1a2=a1+2=1+2=3a3=a2+2=3+2=5a4=a3+2=5+2=7a5=a4+2=7+2=9⋮

In general, given the first term a1 of an arithmetic sequence and its common difference d, we can write the following:

a2=a1+da3=a2+d=(a1+d)+d=a1+2da4=a3+d=(a1+2d)+d=a1+3da5=a4+d=(a1+3d)+d=a1+4d⋮

From this we see that any arithmetic sequence can be written in terms of its first element, common difference, and index as follows:

an=a1+(n−1)d   Arithmetic Sequence

In fact, any general term that is linear in n defines an arithmetic sequence.

Example 1

Find an equation for the general term of the given arithmetic sequence and use it to calculate its 100th term: 7,10,13,16,19,…

Solution:

Begin by finding the common difference,

d=10−7=3

Note that the difference between any two successive terms is 3. The sequence is indeed an arithmetic progression where a1=7 and d=3.

an=a1+(n−1)d=7+(n−1)⋅3=7+3n−3=3n+4

Therefore, we can write the general term an=3n+4. Take a minute to verify that this equation describes the given sequence. Use this equation to find the 100th term:

a100=3(100)+4=304

Answer: an=3n+4; a100=304

The common difference of an arithmetic sequence may be negative.

Example 2

Find an equation for the general term of the given arithmetic sequence and use it to calculate its 75th term: 6,4,2,0,−2,…

Solution:

Begin by finding the common difference,

d=4−6=−2

Next find the formula for the general term, here a1=6 and d=−2.

an=a1+(n−1)d=6+(n−1)⋅(−2)=6−2n+2=8−2n

Therefore, an=8−2n and the 75th term can be calculated as follows:

a75=8−2(75)=8−150=−142

Answer: an=8−2n; a100=−142

The terms between given terms of an arithmetic sequence are called arithmetic meansThe terms between given terms of an arithmetic sequence..

Example 3

Find all terms in between a1=−8 and a7=10 of an arithmetic sequence. In other words, find all arithmetic means between the 1st and 7th terms.

Solution:

Begin by finding the common difference d. In this case, we are given the first and seventh term:

an=a1+(n−1)d  Use n=7.a7=a1+(7−1)da7=a1+6d

Substitute a1=−8 and a7=10 into the above equation and then solve for the common difference d.

10=−8+6d18=6d3=d

Next, use the first term a1=−8 and the common difference d=3 to find an equation for the nth term of the sequence.

an=−8+(n−1)⋅3=−8+3n−3=−11+3n

With an=3n−11, where n is a positive integer, find the missing terms.

a1=3(1)−11=3−11=−8a2=3(2)−11=6−11=−5a3=3(3)−11=9−11=−2a4=3(4)−11=12−11=1a5=3(5)−11=15−11=4a6=3(6)−11=18−11=7     }     arithmetic meansa7=3(7)−11=21−11=10

Answer: −5, −2, 1, 4, 7

In some cases, the first term of an arithmetic sequence may not be given.

Example 4

Find the general term of an arithmetic sequence where a3=−1 and a10=48.

Solution:

To determine a formula for the general term we need a1 and d. A linear system with these as variables can be formed using the given information and an=a1+(n−1)d:

{a3=a1+(3−1)da10=a1+(10−1)d⇒   {−1=a1+2d48=a1+9d  Use a3=−1.  Use a10=48.

Eliminate a1 by multiplying the first equation by −1 and add the result to the second equation.

{−1=a1+2d48=a1+9d        ⇒×(−1)           +  {1=−a1−2d48=  a1+  9d¯                                                                    49=7d7=d                                                                                             

Substitute d=7 into −1=a1+2d to find a1.

−1=a1+2(7)−1=a1+14−15=a1

Next, use the first term a1=−15 and the common difference d=7 to find a formula for the general term.

an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n

Answer: an=7n−22

Try this! Find an equation for the general term of the given arithmetic sequence and use it to calculate its 100th term: 32,2,52,3,72,…

Answer: an=12n+1; a100=51

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