Sigma Notation

Grade 12 · algebra · 64 practice problems · read aloud

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Sigma Notation: The Power of Summation

Sigma notation (∑) is a concise way to represent the sum of a series of terms. It's incredibly useful in algebra and calculus for working with sequences and series, modeling real-world scenarios involving repeated addition, and forming the foundation for integral calculus. 🧮

How to Read and Solve Sigma Notation

A sigma expression is written as: ∑ (expression)

  1. Identify the Index and Bounds: The variable below ∑ (often i, k, or n) is the index of summation. The lower number is the starting value; the upper number is the ending value.
  2. Write Out the Terms: Substitute the index variable into the expression with each integer value from the lower to upper bound.
  3. Sum the Terms: Add all the resulting terms together to find the final sum.

Worked Examples

Example 1: Evaluate ∑ (2k + 1) from k=1 to 4.

  1. k=1: 2(1)+1 = 3
  2. k=2: 2(2)+1 = 5
  3. k=3: 2(3)+1 = 7
  4. k=4: 2(4)+1 = 9
  5. Sum: 3 + 5 + 7 + 9 = 24

Example 2: Evaluate ∑ k² from k=3 to 5.

  1. k=3: (3)² = 9
  2. k=4: (4)² = 16
  3. k=5: (5)² = 25
  4. Sum: 9 + 16 + 25 = 50

Common Mistakes to Avoid

  • Incorrect Number of Terms: The number of terms is (upper bound - lower bound + 1). For ∑ from i=1 to 5, there are 5 terms, not 4.
  • Misapplying the Index: The index variable is a "dummy variable." Its sole purpose is for counting. The final sum is a number, not an expression containing the index.
  • Order of Operations: Remember PEMDAS when evaluating the expression for each term. Square or multiply before adding constants.

Tips & Tricks

  • Know Your Sum Formulas: Memorize these for efficiency:
    ∑ i from i=1 to n = n(n+1)/2
    ∑ i² from i=1 to n = n(n+1)(2n+1)/6
  • Constant Rule: ∑ c from i=1 to n = c * n, where c is a constant.
  • Deconstruct Complex Sums: ∑ (aᵢ + bᵢ) = ∑ aᵢ + ∑ bᵢ. You can split sums apart.

How to Practice

Start by writing out the first 3 and last term to understand the pattern. Then, progress to using sum formulas for larger upper bounds (e.g., n=100). Create your own problems or find practice sets that mix polynomial, constant, and arithmetic series within the sigma notation. Try rewriting simple sums (like 2+4+6+8+10) using sigma notation to build fluency in both directions. 📈

Practice problems

6 of the 64, worked through step by step — try them before opening the answer.

1 Σ(k=1 to 9) (3k - 4) = ?

Hint: This is an arithmetic series. First, find the first term by substituting k=1, and the last term by substituting k=9. Then use the formula for the sum of an arithmetic series: S = n/2 * (first term + last term).

Show the answer

Answer: 99

  1. Identify the number of terms: n = 9.
  2. Find the first term (k=1): 3(1) - 4 = 3 - 4 = -1.
  3. Find the last term (k=9): 3(9) - 4 = 27 - 4 = 23.
  4. Use the sum formula for an arithmetic series: S = n/2 * (first term + last term).
  5. S = 9/2 * (-1 + 23) = 9/2 * 22 = 9 * 11 = 99.

The answer is 99.

2 Σ(k=0 to 5) (5k + 10) = ?

Hint: This is an arithmetic series. Find the first term by substituting k=0 and the last term by substituting k=5. Then use the formula for the sum of an arithmetic series: S = n/2 * (first term + last term).

Show the answer

Answer: 135

  1. Number of terms: n = 5 - 0 + 1 = 6.
  2. First term (k=0): 5(0) + 10 = 0 + 10 = 10.
  3. Last term (k=5): 5(5) + 10 = 25 + 10 = 35.
  4. Sum formula: S = n/2 * (first term + last term).
  5. S = 6/2 * (10 + 35) = 3 * 45 = 135.

The answer is 135.

3 Emma is a financial analyst tracking the monthly profit of a company over the first 5 months of the year. The profit in thousands of dollars for month n is given by P(n) = 25n + 10, where n = 1, 2, 3, 4, 5. Write the total profit for the 5 months using sigma notation and evaluate the sum.

Hint: Think of sigma notation as a way to add up terms that follow a pattern. Identify the starting and ending values of n, and the expression that gives each term. Then evaluate by substituting each n value and adding the results.

Show the answer

Answer: 425

  1. The profit for month n is P(n) = 25n + 10.
  2. We need the sum from n = 1 to n = 5: sum_{n=1}^{5} (25n + 10).
  3. Evaluate term by term: For n = 1: 25(1) + 10 = 25 + 10 = 35 For n = 2: 25(2) + 10 = 50 + 10 = 60 For n = 3: 25(3) + 10 = 75 + 10 = 85 For n = 4: 25(4) + 10 = 100 + 10 = 110 For n = 5: 25(5) + 10 = 125 + 10 = 135
  4. Add the terms: 35 + 60 + 85 + 110 + 135 = 425. The total profit for the first 5 months is 425 thousand dollars.

4 Zoe is analyzing the total profit from a series of weekly sales. The profit in week k is given by (2k + 8) hundred dollars, where k starts at 1 and ends at 3. Write the total profit for the 3 weeks using sigma notation, then evaluate the sum to find the total profit in hundreds of dollars.

Hint: Think about how sigma notation can represent a sum of terms that follow a linear pattern. How can you split the sum into two simpler sums that you know formulas for?

Show the answer

Answer: 36

  1. Identify the series: Σ(k=1 to 3) (2k + 8).
  2. Split the sum: Σ(k=1 to 3) 2k + Σ(k=1 to 3) 8.
  3. Evaluate Σ(k=1 to 3) 2k = 2 * Σ(k=1 to 3) k = 2 * (3(3+1)/2) = 3(3+1).
  4. Evaluate Σ(k=1 to 3) 8 = 8 * 3.
  5. Total = 83 + 3(3+1) = 36. The total profit is 36 hundred dollars.

5 Aroha is arranging triangular tiles in a visual pattern. The first row has 1 tile, the second row has 3 tiles, the third row has 5 tiles, and the pattern continues. She builds a total of 11 rows. Using sigma notation, write the series representing the total number of tiles, then evaluate the sum.

Hint: Think about how each row's number of tiles relates to the row number. This pattern of odd numbers has a well-known closed-form sum formula that depends only on the number of rows.

Show the answer

Answer: 121

  1. Identify the pattern. Row 1 has 1 tile, row 2 has 3 tiles, row 3 has 5 tiles. The number of tiles in row n is 2n - 1. For 11 rows, n goes from 1 to 11.
  2. Write in sigma notation: sum from n=1 to 11 of (2n - 1).
  3. Use the formula for the sum of the first n odd numbers: sum from n=1 to n of (2n - 1) = n^2.
  4. Substitute n = 11: 11^2 = 121.
  5. Therefore, the total number of tiles is 121.

The answer is 121.

6 Olivia is constructing a visual pattern using square tiles. The first row has 5 tiles, the second row has 10 tiles, the third row has 15 tiles, and this pattern continues for 20 rows. Using sigma notation, write the series representing the total number of tiles in this pattern, then evaluate the sum.

Hint: Examine how the number of tiles changes from one row to the next. This is an arithmetic sequence, so focus on the common difference and the formula for the nth term. Sigma notation will help you sum all terms efficiently.

Show the answer

Answer: 1050

  1. Identify the pattern. Row 1 has 5 tiles, row 2 has 10, row 3 has 15. The number of tiles in row n is 5n. Check: row 1: 5(1) = 5, row 2: 5(2) = 10, row 3: 5(3) = 15.
  2. Write the sum using sigma notation. Total tiles = sum from n=1 to 20 of 5n.
  3. Factor out the constant: sum from n=1 to 20 of 5n = 5 times sum from n=1 to 20 of n.
  4. Use the formula for the sum of the first n natural numbers: sum from n=1 to N of n = N(N+1)/2. Here N = 20, so sum = 20(21)/2 = 420/2 = 210.
  5. Multiply by 5: 5 times 210 = 1050.
  6. The total number of tiles is 1050.
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