# arithmetic sequences & series

```Sequences &amp; Series-Calculations
Calculations with Arithmetic Sequences:
Finding the nth term of an Arithmetic Sequence
ππ = ππ + (π − π)π
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 π1 = 15, π = −5
Solution: 15, 10, 5, 0, -5
1
1
6
3
b) Write the first 5 terms when π1 = , π =
π1 =
π2 =
6
1 1
+
6 3
1
1
2
3
π3 = +
π4 =
1 1 5 7 3
Solution: 6 , 2 , 6 , 6 , 2
1
5 1
+
6 3
7
1
6
3
π5 = +
=
1 2
+
6 6
3
2
6
6
= +
=
=
==
5 2
+
6 6
=
7
2
6
6
= +
3 1
=
6 2
5
6
7
6
9
3
6
2
= =
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Sequences &amp; 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
π1 = 0.2
π = πππππππ
3) Use the formula for finding the nth term:
πππ = π. π + (ππ − π)(π. π)
πππ = π. π + (ππ)π. π
πππ = π. π + π. π
ππ = ππ + (π − π)π
plug in the values we know
simplify
πππ = π
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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
π1 = 3
ππ = −65
π = πππππππ
3) Use the formula for finding the nth term: ππ = ππ + (π − π)π
−ππ = π + (π − π)(−π)
plug in the values we know
−ππ = π − ππ + π
distribute the 0.4
−ππ = π − ππ + π
−ππ = π − ππ
solve for n (add 7 to both sides)
−ππ = −ππ
simplify (divide by -4)
ππ = π
2
Sequences &amp; Series-Calculations
Example: Find ππ and d, given: ππ = π. ππ, πππ πππ = π. ππ
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 ππ = ππ + (π − π)π
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
π6 = 1.35
π10 = 2.15
π = πππππππ
Let n = 10, and k = 6
3) Use the formula for finding the nth term using the kth term: ππ = ππ + (π − π)π
π. ππ = π. ππ + (ππ − π)π
plug in the values we know
π. ππ = π. ππ + ππ
subtract 1.35 from both sides
π. π = ππ
simplify (divide by 4)
π. π = π
4) Use the formula for finding the nth term:
just found
ππ = ππ + (π − π)π ,with the d value we
d = 0.2
π10 = 2.15
π. ππ = ππ + (ππ − π)(π. π)
π. ππ = ππ + π(π. π)
π. ππ = ππ + π. π
plug in the values we know
simplify (subtract 1.8 from both sides)
π. ππ = π
3
Sequences &amp; Series-Calculations
Geometric Series-Calculations
http://www.ltcconline.net/greenl/courses/154/seqser/geobinom.htm
Geometric Sequence: (Finding the next number in the sequence)
ππ = ππ ππ−π
Where r is the common ratio
The sum of a Geometric Series:
ππ (π − ππ )
πΊπ =
π−π
4
Sequences &amp; Series-Calculations
Partial Sum of an Arithmetic Sequence:
ππ +ππ
πΊπ = π (
π
π
) ; if ππ ππ  ππππ€π
OR πΊπ = (πππ + (π − π)π)
π