Sums of elementary series — A/L Combined Mathematics

What must I remember about Sums of elementary series?

• Sum of first n natural numbers: ∑_r=1ⁿ r = 1 + 2 + 3 + ... + n = n(n + 1) / 2.
• Sum of squares of first n natural numbers: ∑_r=1ⁿ r² = 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1) / 6.
• Sum of cubes of first n natural numbers: ∑_r=1ⁿ r³ = 1³ + 2³ + 3³ + ... + n³ = [n(n + 1) / 2]² = (∑_r=1ⁿ r)².
• Derivation via telescoping identity: Derive ∑ r² using expansion (r + 1)³ - r³ = 3r² + 3r + 1; summing from r = 1 to n yields (n + 1)³ - 1 = 3 ∑ r² + 3 ∑ r + n.
• Sum of polynomial terms: For general polynomial term ur = a*r³ + b*r² + c*r + d, evaluate sum as Sn = a ∑ r³ + b ∑ r² + c ∑ r + d*n.

Test yourself (1 minute)

1. Sum of odd numbers

   1 + 3 + 5 + ... + (2n - 1) = ∑_r=1ⁿ (2r - 1) = 2[n(n+1)/2] - n = n².

2. Sum of even numbers

   2 + 4 + 6 + ... + 2n = 2 ∑_r=1ⁿ r = n(n + 1).

3. Factorial polynomial representation

   Express r(r + 1) or r(r + 1)(r + 2) directly to use telescoping or standard sums: ∑_r=1ⁿ r(r + 1) = n(n + 1)(n + 2) / 3.

Practise Sums of elementary series questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 20

What should I study first?

What comes next?

Next in the syllabus

Where does Sums of elementary series fit in the syllabus?

Year 1 · Term 3