1. Series and Financial Applications
Arithmetic Series
A series is the sum of a sequence
Example: 2, 5, 8, 11, 14 is a sequence.
is a series.
1, 2, 3, 4,…, 98, 99, 100 is a sequence.
is a series.

2. Finding the sum of a series (Gauss Method)
Let S =
S =
2S =
2S = 100(101)
S = 5050

3. The sum of the first n terms of an arithmetic series.
Sn = t1 + t2 + t3 + t4 + … + tn–1 + tn
Sn = a + (a+d) + (a+2d) + … [a+(n-2)d] + [a+(n-1)d]
Sn = [a+(n-1)d] + [a+(n-2)d] …+ (a+2d) + (a+d)+ a
2Sn = [2a+(n-1)d] + [2a+(n-1)d]+…+ [2a+(n-1)d]
2Sn = n[2a+(n – 1)d]

4. The sum of a series when we know the first term, the common difference and the number of terms.
The sum of a series when we know the first term, the last term and the number of terms.

5. Example 1: find S20 for the series 6 + 10 + 14 + …

6. Example 2: find the sum of the series given the first and last terms.
b) t1 = 5, t20 = –52
a) t1 = 2, t12 = 36
S12 = 228
S20 = – 470

7. Example 3: In an auditorium the last row has 85 seats, the next has 82 seats, the next 79 seats down to the first row which has 7 seats. How many seats are there in the auditorium? 85 82
Sn = … + 7 79
a = 85
d = – 3
n = ? 7
To determine the number of terms we will use the general formula for an arithmetic sequence.
tn = a + (n – 1)d
n = 27
7 = 85 + (n – 1)(–3)

8. a = 85
d = – 3
n = 27
We can now use either formula for the sum of a series.
S27 = 1242

9. tn = a + (n – 1)d n = 6 and t6 = 23 23 = a + 5d 55 = 2a + 11d
Example 4: The sixth term of a series is 23 and the sum of the first 12 terms is 330. Find the first three terms of the series.
tn = a + (n – 1)d
n = 6 and t6 = 23
23 = a + 5d
55 = 2a + 11d
×(–2)
–46 = –2a – 10d
9 = d
t1, t2 t3
– 22, – 13, – 4
a = – 22