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Sequences and Series (3)

Sequences and Series (3). Learn how to calculate THE SUM OF TERMS IN A ARITHMETIC SEQUENCE. The can pyramid…. How many cans are there in this pyramid. How many cans are there in a pyramid with 100 cans on the bottom row?. Sum of Terms.

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Sequences and Series (3)

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  1. Sequences and Series (3) Learn how to calculate THE SUM OF TERMS IN A ARITHMETIC SEQUENCE

  2. The can pyramid… How many cans are there in this pyramid. How many cans are there in a pyramid with 100 cans on the bottom row?

  3. Sum of Terms An Arithmetic series is the sum of the terms in an Arithmetic sequence. Eg. 1, 2, 3, 4… (Arithmetic sequence) 1 + 2 + 3 + 4… (sum of terms)

  4. Back to the pyramid… We wanted to work out the sum of: 1 + 2 + 3 + ….. + 98 + 99 + 100 If we write it out in reverse we get…. 100 + 99 + 98 + ….. + 3 + 2 + 1 101 + 101 + 101 +….. How many times do we add 101 together? 101 x 100 = 10100 What do we need to do to this answer? 10100 / 2 = 5050

  5. Activity 1 Work out the sum of the first 50 positive integers. Work out the sum of all the odd numbers from 21 up to 99.

  6. a + (a + d) + (a + 2d) + (a + 3d) + …. + (l - 3d) + (l - 2d) + (l - d) + l l + (l - d) + (l - 2d) + (l - 3d) + …. + (a + 3d) + (a + 2d) + (a + d) + a Sum of Terms Work out the sum of all the odd numbers from 21 up to 99. a = first term l = last term d = common difference There are n pairs of numbers that add up to (a + l) Sum = ½ n (a + l) Sum = ½ n (u1 + un)

  7. Sum of Terms u1 = first term un = last term d = common difference Sum of Terms Sn = ½ n (u1 + un) From last lesson, we know that the nth term (last term) is given by: un = u1 + (n – 1) d Sn = ½ n (2u1 + (n – 1) d)

  8. Sum of Terms (Sn) u1 = first term un = last term d = common difference Arithmetic Series = ½ n (u1 + un) Arithmetic Series = ½ n (2u1 + (n – 1) d) Why are both of these formulae useful?

  9. Example 1 Find the sum of the arithmetic series:11 + 15 + 19 + … + 107 u1 = 11 d = 4 un = 107 l = u1 + (n – 1)d From last lesson… 107 = 11 + 4(n – 1) Sub values in… n = 25 Solving… Sum = ½ n (u1 + un) Using formula… Sum = ½ 25 (11 + 107) Sub values in… Sum = 1475

  10. Activity Turn to page 186-7 of your textbook and answer questions 6-12 in exercise 6C

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