Read the lower limit FIRST โ that’s your starting value of r. Step it up one at a time until you hit the upper limit, substituting each into the expression.
Evaluating a sigma sum
Two methods:
By hand โ list out each term and add. Works for any short sum.
On the GDC โ most calculators have a ฮฃ function. Type the expression, the variable, the lower limit, and the upper limit. Useful for long sums or to check by hand.
GDC tip: on TI-Nspire / Casio Classpad, the ฮฃ template is under the maths/template menu. Always check the lower limit on the GDC matches the lower limit in the question.
๐งญ Recipe โ work with any sigma sum
Identify lower limit, upper limit, and expression.
Count the terms: number of terms = upper โ lower + 1.
Substitute each value of the index into the expression.
Add all the resulting terms.
Use the GDC’s ฮฃ to verify, especially for sums with 6+ terms.
Lower limit is 3, NOT 1 โ start at k = 3k=3: 3ยฒ = 9k=4: 4ยฒ = 16k=5: 5ยฒ = 25k=6: 6ยฒ = 36Add9 + 16 + 25 + 36 = 86= 86always read the lower limit โ starting at k=1 here would have given the wrong sum.
Spot the patterneach term = 5 ร (its position)expression: 5nSet the limits6 terms โ n goes from 1 to 6ฮฃ(n=1 to 6) 5ncheck: 5(1)+5(2)+…+5(6) = 5+10+15+20+25+30 โ
WE 4
Convert un formula to sigma and evaluate
A sequence has un = 3n โ 2. Write u1 + u2 + โฆ + u8 in sigma notation, then evaluate.
Write in sigma formฮฃ(n=1 to 8) (3n โ 2)Substitute and addterms: 1, 4, 7, 10, 13, 16, 19, 22sum = 1+4+7+10+13+16+19+22ฮฃ(n=1 to 8)(3nโ2) = 92
WE 5
Long sum โ use the GDC
Use your GDC to evaluate โ20r=1 (rยฒ + 1).
Enter on GDC’s ฮฃ templateexpression: rยฒ + 1variable: rlower: 1, upper: 20GDC returns= 289020 terms by hand would take ages โ let the GDC do it.
WE 6
Real-world โ savings using sigma
A savings plan adds $(3n โ 1) to an account in week n. Use sigma notation to find the total saved over the first 10 weeks.
Set up sigmatotal = ฮฃ(n=1 to 10) (3n โ 1)Evaluate (GDC or by hand)terms: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29sum = 155total = $155sigma is just shorthand โ once you can set it up, the GDC does the adding.
๐ก Top tips
Always read the lower limit โ it’s not always 1.
Number of terms = upper โ lower + 1 โ use this to sanity-check.
The letter is arbitrary: r, k, n โ all give the same sum if the limits and expression match.
Use the GDC’s ฮฃ for any sum with more than ~6 terms โ saves time and avoids arithmetic slips.
โ Common mistakes
Starting at the wrong index: โk=3 means start at k = 3, not k = 1.
Miscounting terms: โ from 7 to 14 has 8 terms, not 7. (14 โ 7 + 1 = 8.)
Forgetting brackets in the expression: โ(2r + 1) is different from โ2r + 1 โ the first sums (2r+1), the second sums 2r then adds 1 once.
Substituting the limit value once instead of every index: you have to substitute EACH value of r from lower to upper.
Up next: Arithmetic Sequences & Series โ when terms go up (or down) by a fixed amount each time. Comes with two ready-made formulae so you don’t have to add term-by-term.
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