Sums of Powers of Natural Numbers
Standard sums give 1 + 2 + ... + n = n(n+1)/2, the sum of squares n(n+1)(2n+1)/6 and the sum of cubes the square of the first sum.
What is Sums of Powers of Natural Numbers?
Standard sums give 1 + 2 + ... + n = n(n+1)/2, the sum of squares n(n+1)(2n+1)/6 and the sum of cubes the square of the first sum.
Key formula / rule: Split the general term into standard power sums before summing.
Key points
- Derive the sum of the first n natural numbers by pairing.
- Compute the sum of a series by decomposing its general term.
Common exam trap
Summing a general term written for the wrong index range.
Definitions
- Term
Sums of Powers of Natural Numbers
- Meaning
Standard sums give 1 + 2 + ... + n = n(n+1)/2, the sum of squares n(n+1)(2n+1)/6 and the sum of cubes the square of the first sum.
- Term
Sums of Powers of Natural Numbers — explanation
- Meaning
Once a general term is decomposed into powers of n, these three results evaluate almost any polynomial series. Telescoping handles the rational cases that remain.
Learning objectives
Derive the sum of the first n natural numbers by pairing.
Compute the sum of a series by decomposing its general term.
Formulae
- Key point
Split the general term into standard power sums before summing.
- Key point
The sum of the first n cubes equals the square of the sum of the first n numbers.
- Key point
Telescoping sums cancel all interior terms, leaving only the ends.
Prerequisites
MAT-U1-SS-T1-S1-C1
Common mistakes
Summing a general term written for the wrong index range.
Applying a power-sum formula to a term that is not a polynomial in n.
Keywords
Sums
Powers
Natural
Numbers
Practice preview
What is the sum of the cubes of the first 4 natural numbers?…
easy
Find the sum of the series 1*2 + 2*3 + 3*4 + ... + 10*11.…
medium
Calculate the sum of the series Σ (k² - k) for k from 1 to 8.…
medium