7.5 Arithmetic Series

Homefun  pg 452-453 # 1ac, 3, 4be, 5acd, 6bf, 7cd, 8, 14, 15

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

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 = 2 describes 2, 6, 18, 54, … tn =3 tn-1 t2 =3t1 =3(2) = 6 n>1 , n  N t3 =3t2 =3(6) = 18

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

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 = 4 11 – 7 = 4 15 – 11 = 4

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

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

GAUSS’ METHOD When German mathematician Karl Friedrich Gauss (177721855) 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.

GAUSS’ Method Consider the arithmetic series 1 + 6 + 11 + 16 + 21 + 26 + 31 + 36. 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.

GAUSS’ Method Consider the arithmetic series 1 + 6 + 11 + 16 + 21 + 26 + 31 + 36. 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.

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 = 2 + 10 + 18 +26 S4 = 56

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 = 2 + 10 + 18 +26 S4 = 56

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

Derivation of the formulA

Derivation of the formula

ExampleS Determine the sum of the first 25 terms of the series – 5 – 8 – 11 – … Determine the sum of the series 16 + 10 + 4 + … – 50 Given S36 = – 540 and d = 4 for an arithmetic series, find t10.

Example 4 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?

Example 4 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?

Example 4 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?

Homefun  pg 452-453 # 1ac, 3, 4be, 5acd, 6bf, 7cd, 8, 14, 15