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

Finding the sum of a series (Gauss Method) Let S = 1 + 2 + 3 + ... + 99 + 100 S = 100 + 99 + 98 + ... + 2 + 1 2S = 101 +101 +101 + ... +101 + 101 2S = 100(101) S = 5050

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]

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.

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

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

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 = 85 + 82 + 79 + … + 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)

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

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