Sequences & Series-Calculations
Calculations with Arithmetic Sequences:
Finding the nth term of an Arithmetic Sequence
𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅
Identifying an arithmetic sequence:
Example: 2, 5, 8, 11, 14
Process: Pick 2 pairs of consecutive numbers and calculate the common difference
8–5=3
11 - 8 = 3
Solution: If the answers are the same, then it’s an arithmetic sequence with d = 3
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Writing the terms of an arithmetic sequence:
Examples:
a) Write the first 5 terms when 𝑎1 = 15, 𝑑 = −5
Solution: 15, 10, 5, 0, -5
1
1
6
3
b) Write the first 5 terms when 𝑎1 = , 𝑑 =
𝑎1 =
𝑎2 =
6
1 1
+
6 3
1
1
2
3
𝑎3 = +
𝑎4 =
1 1 5 7 3
Solution: 6 , 2 , 6 , 6 , 2
1
5 1
+
6 3
7
1
6
3
𝑎5 = +
=
1 2
+
6 6
3
2
6
6
= +
=
=
==
5 2
+
6 6
=
7
2
6
6
= +
3 1
=
6 2
Copyright © 2011 Lynda Aguirre
5
6
7
6
9
3
6
2
= =
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Sequences & Series-Calculations
Example: Find the 21st term of 0.2, 0.6, 1.0, 1.4, 1.8, . . .
1) Is this an arithmetic sequence?
Find d (pick two consecutive pairs and see if d is the same #)
d = 0.6 – 0.2 = 0.4 ; d = 1.0 – 0.6 = 0.4 → it has the same d, so it’s arithmetic
2) What information do we have now?
d = 0.4
𝑎1 = 0.2
𝑛 = 𝒖𝒏𝒌𝒏𝒐𝒘𝒏
3) Use the formula for finding the nth term:
𝒂𝟐𝟏 = 𝟎. 𝟐 + (𝟐𝟏 − 𝟏)(𝟎. 𝟒)
𝒂𝟐𝟏 = 𝟎. 𝟔 + (𝟐𝟎)𝟎. 𝟒
𝒂𝟐𝟏 = 𝟎. 𝟔 + 𝟖. 𝟒
𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅
plug in the values we know
simplify
𝒂𝟐𝟏 = 𝟗
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Example: Find the number of terms given: 3, -1, -5, -9, . . ., -65
1) Is this an arithmetic sequence?
Find d (pick two consecutive pairs and see if d is the same #)
d = -1 – 3 = -4 ; d = -5 – (-1) = -4 → it has the same d, so it’s arithmetic
2) What information do we have now?
d = -4
𝑎1 = 3
𝑎𝑛 = −65
𝑛 = 𝒖𝒏𝒌𝒏𝒐𝒘𝒏
3) Use the formula for finding the nth term: 𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅
−𝟔𝟓 = 𝟑 + (𝒏 − 𝟏)(−𝟒)
plug in the values we know
−𝟔𝟓 = 𝟑 − 𝟒𝒏 + 𝟒
distribute the 0.4
−𝟔𝟓 = 𝟑 − 𝟒𝒏 + 𝟒
add like terms
−𝟔𝟓 = 𝟕 − 𝟒𝒏
solve for n (add 7 to both sides)
−𝟓𝟔 = −𝟒𝒏
simplify (divide by -4)
𝟏𝟒 = 𝒏
Copyright © 2011 Lynda Aguirre
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Sequences & Series-Calculations
Example: Find 𝒂𝟏 and d, given: 𝒂𝟔 = 𝟏. 𝟑𝟓, 𝒂𝒏𝒅 𝒂𝟏𝟎 = 𝟐. 𝟏𝟓
The problem is that the formula we have for finding the nth term requires consecutive terms,
the 6th
and 10th term are not
consecutive. So we
Formula for the nth term, given a random kth term
have
to alter our formula
as
follows:
Since n=k+ (n-k), we can say that 𝒂𝒏 = 𝒂𝒌 + (𝒏 − 𝒌)𝒅
2) Is this an arithmetic sequence?
We were given the value for the common difference, d, so we assume it’s arithmetic
2) What information do we have now?
d = unknown
𝑎6 = 1.35
𝑎10 = 2.15
𝑛 = 𝒖𝒏𝒌𝒏𝒐𝒘𝒏
Let n = 10, and k = 6
3) Use the formula for finding the nth term using the kth term: 𝒂𝒏 = 𝒂𝒌 + (𝒏 − 𝒌)𝒅
𝟐. 𝟏𝟓 = 𝟏. 𝟑𝟓 + (𝟏𝟎 − 𝟔)𝒅
plug in the values we know
𝟐. 𝟏𝟓 = 𝟏. 𝟑𝟓 + 𝟒𝒅
subtract 1.35 from both sides
𝟎. 𝟖 = 𝟒𝒅
simplify (divide by 4)
𝟎. 𝟐 = 𝒅
4) Use the formula for finding the nth term:
just found
𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅 ,with the d value we
d = 0.2
𝑎10 = 2.15
𝟐. 𝟏𝟓 = 𝒂𝟏 + (𝟏𝟎 − 𝟏)(𝟎. 𝟐)
𝟐. 𝟏𝟓 = 𝒂𝟏 + 𝟗(𝟎. 𝟐)
𝟐. 𝟏𝟓 = 𝒂𝟏 + 𝟏. 𝟖
plug in the values we know
simplify (subtract 1.8 from both sides)
𝟎. 𝟑𝟓 = 𝒅
Copyright © 2011 Lynda Aguirre
3
Sequences & Series-Calculations
Geometric Series-Calculations
http://www.ltcconline.net/greenl/courses/154/seqser/geobinom.htm
Geometric Sequence: (Finding the next number in the sequence)
𝒂𝒏 = 𝒂𝟏 𝒓𝒏−𝟏
Where r is the common ratio
The sum of a Geometric Series:
𝒂𝟏 (𝟏 − 𝒓𝒏 )
𝑺𝒏 =
𝟏−𝒓
Copyright © 2011 Lynda Aguirre
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Sequences & Series-Calculations
Partial Sum of an Arithmetic Sequence:
𝒂𝟏 +𝒂𝒏
𝑺𝒏 = 𝒏 (
𝒏
𝟐
) ; if 𝑎𝑛 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛
OR 𝑺𝒏 = (𝟐𝒂𝟏 + (𝒏 − 𝟏)𝒅)
𝟐
Copyright © 2011 Lynda Aguirre
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