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Aim: What is this symbol It’s Greek to me!

Aim: What is this symbol It’s Greek to me!. Do Now:. Find the sum of the geometric series. . The sum of the first n terms of a sequence is represented by. where i is the index of summation,. n is the upper limit of summation, and. 1 is the lower limit of summation.

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Aim: What is this symbol It’s Greek to me!

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  1. Aim: What is this symbol It’s Greek to me! Do Now: Find the sum of the geometric series.

  2. The sum of the first n terms of a sequence is represented by where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. Definition of Summation Notation sum of terms sigma When n is a specific number, the sum of a sequence is called a finite series.

  3. where i is the index of summation 0 is the lower limit of summation n is the upper limit of summation Summation Notation The first term of the summation is formed by substituting the lower limit for the index into the general term. Each succeeding term of the summation is formed using successive integral values of the index, until the upper limit is reached.

  4. Properties of Sums

  5. Model Problems Find the value of each summation = 2(0 + 1 + 2 + 3 + 4 + 5 + 6) = 2(21) = 42 = (2 – 2)2 + (3 – 2)2 + (4 – 2)2 + (5 – 2)2 = 02 + 12 + 22 + 32 = 14

  6. Model Problems Rewrite using the summation symbol 2(1) + 2(2) + 2(3) + 2(4) + 2(5)

  7. Regents Questions = 2

  8. The Sum of an Arithmetic Sequence: Series In an arithmetic series, if a1 is the first term, n is the number of terms, an is the nth term, and d is the common difference, then Sn the sum of the arithmetic series, is given by the formulas: or

  9. The nth Term of an Arithmetic Sequence The nth term of an arithmetic sequence has the form an = dn + c where d is the common difference between consecutive terms of the sequence and c = a1 – d An alternative form of the nth term is an = a1 + (n – 1)d Summation and Arithmetic Series

  10. Regents Questions Find the partial sum of the following arithmetic series = 115

  11. The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a1,a1r2, a1r3,a1r4, . . . . a1rn - 1 . . . . with common ratio r 1 is given by Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .

  12. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . . a1a2a3a4 . . . . . an . . . . a1a1ra1r2a1r3 . . . . a1rn - 1 . . . . The sum of the finite geometric sequence The Sum of a Finite Geometric Sequence r = ? 3 a1 + a1r + a1r2 +a1r3 + . . . + a1r8 - 1 =

  13. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . . The Sum of a Finite Geometric Sequence r = ? 3 = 1(3)7 1 + 3 + 9 + 27 +. . .+ = 3280 1 + 1(3) + 1(3)2 + 1(3)3 + . . . . = 3280

  14. Find the sum of Model Problem a1 = 10; r = -1/2 ; n = 11 How many terms in the series? starting at i = 0, there are 11 terms

  15. Regents Questions Find the sum of the following geometric series. = 4368

  16. Find The Sum of a Infinite Geometric Sequence If |r| < 1, then the infinite geometric sequence a1,a1r2, a1r3,a1r4, . . . . a1rn - 1 . . . has the sum = 4 + 4(0.6) + 4(0.6)2 + 4(0.6)3 + . . a1 = 4 and r = 0.6 - (|r| < 1)

  17. Model Problem Find the sum of 3 + 0.3 + 0.03 + 0.003 + . . . a1 = 3 and r = ? 0.1 3 + 3(0.1) + 3(0.1)2 + 3(0.1)3 + . . .

  18. Model Problem Express the series in sigma notation and find the sum. 54 + 18 + 6 + 2 + 2/3 + 2/9 728/9

  19. Model Problem Find the sum of a1 = 4(0.3)1 = 1.2 r = 0.3 n = 12  1.714

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