Sum of a Linear Number Sequence Calculator
Our free 3d geometry calculator solves sum alinear number sequence problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
S = sum of (a + i*d)^p for i = 0 to n-1
Where a is the first term, d is the common difference, n is the number of terms, and p is the power applied to each term. For p=1 this reduces to the standard arithmetic series sum. For p=2, the sum-of-squares formula applies.
Worked Examples
Example 1: Sum of Squares of Arithmetic Sequence
Problem:Find the sum of squares of the first 5 terms of the sequence 3, 7, 11, 15, 19 (a=3, d=4, n=5, p=2).
Solution:Terms: 3, 7, 11, 15, 19\nSquares: 9, 49, 121, 225, 361\nSum of squares = 9 + 49 + 121 + 225 + 361 = 765\nVerification with formula: n*a^2 + n(n-1)*a*d + d^2*n(n-1)(2n-1)/6\n= 5(9) + 5(4)(3)(4) + 16(5)(4)(9)/6\n= 45 + 240 + 480 = 765
Result:Sum of squares = 765 | Linear sum = 55 | Mean squared term = 153
Example 2: Sum of Cubes of Natural Numbers
Problem:Calculate the sum of cubes of the first 6 natural numbers (a=1, d=1, n=6, p=3).
Solution:Terms: 1, 2, 3, 4, 5, 6\nCubes: 1, 8, 27, 64, 125, 216\nSum of cubes = 1 + 8 + 27 + 64 + 125 + 216 = 441\nVerification: [n(n+1)/2]^2 = [6(7)/2]^2 = 21^2 = 441
Result:Sum of cubes = 441 = 21^2 | Linear sum = 21 | Confirms Nicomachus theorem
Frequently Asked Questions
What is an alinear number sequence and how does it differ from a linear one?
An alinear (or nonlinear) number sequence is formed by applying a nonlinear transformation, such as raising to a power, to the terms of an arithmetic (linear) sequence. In a standard linear arithmetic sequence like 2, 5, 8, 11, 14, each term increases by a constant difference. An alinear transformation might square each term to produce 4, 25, 64, 121, 196, where the differences between consecutive terms are no longer constant. The sum of such transformed sequences cannot be computed using the simple arithmetic series formula n/2 times (first + last) and instead requires either direct computation or specialized formulas depending on the power applied.
How do you calculate the sum of squares of an arithmetic sequence?
For an arithmetic sequence with first term a, common difference d, and n terms, the sum of squares has a closed-form formula. It equals n times a squared, plus n times (n-1) times a times d, plus d squared times n times (n-1) times (2n-1) divided by 6. This formula derives from expanding (a + id)^2 = a^2 + 2aid + i^2d^2 and summing each part separately using known formulas for sum of integers and sum of squares of integers. For the basic sequence 1, 2, 3, ..., n (where a=1 and d=1), the sum of squares simplifies to the well-known formula n(n+1)(2n+1)/6. This result has applications in statistics for computing variance.
What is the formula for the sum of an arithmetic series?
The sum of a standard arithmetic series with first term a, common difference d, and n terms is S = n/2 times (2a + (n-1)d), which can also be written as S = n/2 times (first term + last term). This formula was famously discovered by young Carl Friedrich Gauss, who reportedly summed the numbers 1 to 100 by pairing 1 with 100, 2 with 99, and so on, yielding 50 pairs each summing to 101, for a total of 5050. The formula works because the average of an arithmetic sequence is exactly the average of its first and last terms. This linear sum serves as the baseline from which alinear sums deviate.
What are Faulhaber formulas and how do they relate to alinear sums?
Faulhaber formulas provide closed-form expressions for the sum of the p-th powers of the first n natural numbers: 1^p + 2^p + 3^p + ... + n^p. For p=1, the sum is n(n+1)/2. For p=2, it is n(n+1)(2n+1)/6. For p=3, it is [n(n+1)/2]^2, which is remarkably the square of the sum of the first n numbers. For higher powers, the formulas involve Bernoulli numbers and become increasingly complex. These formulas can be adapted to arithmetic sequences with arbitrary first term and common difference by substituting the general term a + id for each i. Faulhaber published these results in 1631, and they remain important in number theory and combinatorics.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy