Formulas that must be memorized:
- Σ_{k=1}^{n} 1 = n
- Σ_{k=1}^{n} k = n(n+1)/2
- Σ_{k=1}^{n} k² = n(n+1)(2n+1)/6
- Σ_{k=1}^{n} k³ = {n(n+1)/2}²
Example ① Find Σ_{k=1}^{10} k².
Solution:10·11·21/6 = 2310/6 = 385
Solution:10·11·21/6 = 2310/6 = 385
Example ② Express Σ_{k=1}^{n} (3k² − 2k + 1) in terms of n.
Solution:3·n(n+1)(2n+1)/6 − 2·n(n+1)/2 + n
= n(n+1)(2n+1)/2 − n(n+1) + n
Solution:3·n(n+1)(2n+1)/6 − 2·n(n+1)/2 + n
= n(n+1)(2n+1)/2 − n(n+1) + n
Key Points
- Memorize all four formulas completely
- For the sum of a polynomial, decompose it into individual terms and apply the formulas
Practice Problem
- Σ_{k=1}^{20} k = ?
Answer and Explanation
- Answer:210
20·21/2=210.