Skip to content

Mathematics · Ch 5 — Binomial Theorem, Sequences and Series

Some Special Finite Series

5.5.3

Some Special Finite Series

Three specific formulas for summing finitely many terms recur constantly enough to be worth stating on their own (rather than re-derived via the AP/AGP machinery each time):

1. Sum of the first nn natural numbers:

∑k=1nk=1+2+3+⋯+n=n(n+1)2\sum_{k=1}^n k = 1+2+3+\cdots+n = \frac{n(n+1)}2

(this is just the AP-sum formula with a=d=1a=d=1).

2. Sum of the squares of the first nn natural numbers:

∑k=1nk2=12+22+32+⋯+n2=n(n+1)(2n+1)6\sum_{k=1}^n k^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}6

(derivable from the identity a3−b3=(a−b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2) applied telescopically).

3. Sum of the cubes of the first nn natural numbers:

∑k=1nk3=13+23+33+⋯+n3=(n(n+1)2)2\sum_{k=1}^n k^3 = 1^3+2^3+3^3+\cdots+n^3 = \left(\frac{n(n+1)}2\right)^2

(derivable from the identity k4−(k−1)4=4k3−6k2+4k−1k^4-(k-1)^4=4k^3-6k^2+4k-1 applied telescopically, using the first two formulas along the way). …

Start your 14-day free trial to unlock the full solution →
← Previous · 5.5.2Q12 — In a certain town, a viral disease caused severe health hazards upon i…Next · 5.6 →Infinite Sequences and Series