Sigma notation

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What this lesson covers

  • The numbers on sigma give the first and last value of k.
  • Writing out a few terms shows the type of series.
  • A written-out series can be packed back into sigma.
Key idea

Reading the symbol

  • Σ with k = 1 underneath and 5 on top means use k = 1, 2, 3, 4, 5.
  • Put each value into the rule, then add the results.
Worked example

Σ (2k + 1), k = 1 to 5

  • Write the terms out.
  1. 3 + 5 + 7 + 9 + 11
  2. = 35
Worked example

Σ (4k − 1), k = 1 to 50

  • The first few terms show that it is arithmetic.
  1. 3 + 7 + 11 + ...
  2. a = 3, last term = 199, n = 50
  3. S = 25(3 + 199)
  4. S = 5050
Worked example

Write 2 + 5 + 8 + ... + 29 in sigma

  • Find the rule and the number of terms.
  1. difference 3, so the rule is 3k − 1
  2. 3k − 1 = 29, so k = 10
  3. Σ (3k − 1), k = 1 to 10

Common mistakes

  • Adding the top and bottom numbers.
    They are limits for k, not values in the sum.
  • Treating a sum that starts at k = 2 as having the same count.
    Count the terms: top − bottom + 1.

Your turn

  • 1. Find Σ k², k = 1 to 6.
  • 2. Find Σ (2k + 3), k = 1 to 30.
  • 3. Write 5 + 10 + 20 + 40 + 80 in sigma.
Show answer

1. 91 2. 15(5 + 63) = 1020 3. Σ 5 × 2ᵏ⁻¹, k = 1 to 5

Summary

What to take away

  • Sigma is an instruction to add.
  • Write a few terms before choosing a formula.

More in AS 91258

Arithmetic sequences and seriesNotesGeometric sequences and seriesNotesSigma notationNotesGrowth and financial problemsNotes

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