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Welcome to our Math lesson on The Combination of Sequences and Series, this is the fifth lesson of our suite of math lessons covering the topic of Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series., you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
How to Combine Sequences and Series
In some questions, there are situations where there is missing information, which is completed by combining the sequences and series approach. Let's consider a couple of examples - one from each type - to clarify this point.
Example 6
In an arithmetic progression, the first term is -4, the fifth term is 20 and the last term is 134. Calculate the sum of all terms in the corresponding series.
Solution 6
We have the following clues: x1 = -4, x5 = 20 and xn = 134. Thus, since in an arithmetic series the nth term is calculated by the formula
xn = x1 + (n - 1) ∙ d
we first find the common difference d by writing
x5 = x1 + (5 - 1) ∙ d
20 = -4 + 4 ∙ d
20 + 4 = 4 ∙ d
24 = 4d
d = 6
Now, let's find the total number of terms n in this sequence. Thus,
xn = x1 + (n - 1) ∙ d
134 = -4 + (n - 1) ∙ 6
n - 1 = (34 + 4/6
n - 1 = 23
n = 23 + 1
n = 24
At this point, we are ready to find the sum of all terms in this sequence, which represents the corresponding arithmetic series. Applying the Gauss formula, we have
Sn = (x1 + xn ) ∙ n/2
S24 = (x1 + x24 ) ∙ 24/2
S24 = (-4 + 134) ∙ 24/2
= 130 ∙ 12
= 1560
Let's consider another example, but this time with geometric series.
Example 7
The second term of a geometric sequence is 32 times greater than the seventh one. The sum of the first seven terms of this sequence is 254. What is the 13th term of this geometric sequence?
Solution 7
We have the following clues: x2 = 32x7 and S7 = 254. These clues help us find the common ratio R first. Thus, since in a geometric sequence xn = Rn - 1 · x1, we have
x7 = R7 - 1 ∙ x1
and
x2 = R2 - 1 ∙ x1
More precisely, we have
x7 = R6 ∙ x1
and
x2 = R ∙ x1
Dividing the above equations yields
x7/x2 = R6 ∙ x1/R ∙ x1
Simplifying x1 yields
x7/x2 = R6/R
x7/x2 = R6 - 1
x7/x2 = R5
Inverting down both sides yields
x2/x7 = 1/R5
Since x2 = 32x7, then
x7 = x2/32
Thus,
x2/x2/32 = 1/R5
1/R5 = 32
R5 = 1/32
R5 = (1/2)5
R = 1/2
Next, let's find the first term x1 of this sequence by the help of the geometric series formula (the second version, given that R < 1). Thus, since in a geometric series
Sn = x1 (1 - Rn)/1 - R
and given that in the specific case S7 = 254 (n = 7), we obtain for the first term x1
254 = x1 [1 - (1/2)7 ]/1 - 1/2
254 = x1 [1 - 17/27 ]/2/2 - 1/2
254 = x1 [1 - 1/128]/1/2
254 ∙ 1/2 = x1 (1 - 1/128)
127 = x1 (128/128 - 1/128)
127 = x1 ∙ 127/128
x1 = 128
Last, we find the 13th term of the given geometric sequence by using again the formula
xn = Rn - 1 ∙ x1
but this time for n = 13. Hence, substituting the known values yields
x13 = (1/2)13 - 1 ∙ 128
= (1/2)12 ∙ 128
= 2 - 12 ∙ 27
= 2 - 12 + 7
= 2-5
= 1/25
= 1/32
More Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. Lessons and Learning Resources
Sequences and Series Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 12.2 | Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 12.2.1 | Series versus Sequences | | |
| 12.2.2 | The Gauss Method and Arithmetic Series | | |
| 12.2.3 | An Alternative Formula for the Calculation of Sn in Arithmetic Series | | |
| 12.2.4 | Geometric Series | | |
| 12.2.5 | The Combination of Sequences and Series | | |
| 12.2.6 | Combined Series | | |
| 12.2.7 | The Practical Applications of Number Series and Sequences | | |
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