1. 7.5 Arithmetic Series

2. Homefun 
pg # 1ac, 3, 4be, 5acd, 6bf, 7cd, 8, 14, 15

3. REVIEW Sequence: an ordered list of numbers
Term: a number in a sequence (the first term is referred to as t1, the second term as t2, etc…)
example
3, 7, 11, 15, …
t1 = 3 t2 = 7 t3 = 11 t4 = 15

4. REVIEW - Recursive SEQUENCE
a sequence for which one or more terms are given
each successive term is determined by performing a calculation using the previous term(s)
example
t1 = describes 2, 6, 18, 54, …
tn =3 tn t2 =3t1 =3(2) = 6
n>1 , n  N t3 =3t2 =3(6) = 18

5. REVIEW - General term
a formula that expresses each term of a sequence as a function of its position
labelled tn
example
tn = 2n describes 2, 4, 6, 8, 10

6. REVIEW - arithmetic SEQUENCE
a sequence that has a common difference between any pair of consecutive terms
The general arithmetic sequence is a, a + d, a + 2d, a + 3d, …, where a is the first term and d is the common difference.
example
3, 7, 11, 15, … has a common difference of 4
7 – 3 = – 7 = – 11 = 4

7. REVIEW - Arithmetic sequence
General term
Recursive formula
tn = a + (n – 1)d
where
a is the first term
d is the common difference
n  N
t1 = a
tn = tn-1 + d
n > 1 , n  N

8. REVIEW - Arithmetic sequence
DISCRETE Linear function
f (n) = dn + b
where b = t0 = a - d

9. GAUSS’ METHOD
When German mathematician Karl Friedrich Gauss ( ) was a child, his teacher asked him to calculate the sum of the numbers from 1 to 100.
Gauss wrote the list of numbers twice, once forward and once backward.
He then paired terms from the
two lists to solve the problem.

10. GAUSS’ Method
Consider the arithmetic series
Use Gauss’s method to determine the sum of this series.
Do you think this method will work for any arithmetic series? Justify your answer.

11. GAUSS’ Method
Consider the arithmetic series
Use Gauss’s method to determine the sum of this series.
Do you think this method will work for any arithmetic series? Justify your answer.

12. seRIES a series is the sum of the terms of a sequence
Sn represents the partial sum of the first n terms of a sequence
example
For the sequence 2, 10, 18, 26, 34, 42, ….
S4 =
S4 = 56

13. Arithmetic seRIES sum of terms of an arithmetic sequence
Sn represents the partial sum of the first n terms of a sequence.
example
For the sequence 2, 10, 18, 26, 34, 42, ….
S4 =
S4 = 56

14. Arithmetic seRIES n > 1 , n  N where a is the first term
d is the common difference
n > 1 , n  N

15. Derivation of the formulA

16. Derivation of the formula

17. ExampleS
Determine the sum of the first 25 terms of the series – 5 – 8 – 11 – …
Determine the sum of the series … – 50
Given S36 = – 540 and d = 4 for an arithmetic series, find t10.

18. Example pg450
In an amphitheatre, seats are arranged in 50 semicircular rows facing a domed stage.
The first row contains 23 seats, and each row contains 4 more seats than the previous row.
How many seats are in the amphitheatre?

19. Example pg450
In an amphitheatre, seats are arranged in 50 semicircular rows facing a domed stage.
The first row contains 23 seats, and each row contains 4 more seats than the previous row.
How many seats are there?

20. Example pg450
In an amphitheatre, seats are arranged in 50 semicircular rows facing a domed stage.
The first row contains 23 seats, and each row contains 4 more seats than the previous row.
How many seats are there?

23. Homefun 
pg # 1ac, 3, 4be, 5acd, 6bf, 7cd, 8, 14, 15