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Sum of a Linear Number Sequence Calculator

Calculate the sum of a linear (first-degree) sequence a(n) = an + b from n = 1 to N.

By Christopher FloiedPublished Updated

The multiplier of n in the formula an + b

The constant added to each term

Sum from n=1 to N

Results

Sum

175

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Sum of a Linear Number Sequence Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Sum of a Linear Number Sequence Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Sum of a Linear Number Sequence Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Sum of a Linear Number Sequence Calculator

A linear number sequence has the form a(n) = an + b, producing terms like a + b, 2a + b, 3a + b, and so on. This is closely related to arithmetic sequences and represents any first-degree polynomial evaluated at consecutive integers. The sum of such a sequence from n = 1 to N can be computed in closed form by separating the sum into a × Σn + b × N = a × N(N+1)/2 + bN. Linear sequences appear in everyday contexts — counting by threes, stacking objects, calculating costs with a fixed base plus a per-unit charge, and analyzing linear complexity algorithms. This calculator gives you the total sum instantly for any linear rule and range.

The Math Behind It

Given a(n) = an + b, the sum S = Σ_{n=1}^{N} (an + b) = a Σn + bN = a N(N+1)/2 + bN. This is equivalent to the arithmetic series formula with first term (a + b), last term (aN + b), and N terms: S = N/2 × (first + last) = N/2 × (a + b + aN + b) = N/2 × (a(N+1) + 2b) = a N(N+1)/2 + bN. The result is a quadratic function of N, reflecting the fact that summing a linear function produces a quadratic. This is the discrete analog of integrating a linear function: ∫₁ᴺ (ax + b)dx = aN²/2 + bN − a/2 − b, which closely matches the discrete sum for large N.

Formula Reference

Sum of Linear Sequence

Σ(an + b) = a × N(N+1)/2 + bN

Variables: a = coefficient, b = constant, N = number of terms

Worked Examples

Example 1: Sum of 3n + 1 for n = 1 to 10

Find Σ(3n + 1) from n=1 to 10.

Step 1:Sum = 3 × 10(11)/2 + 1 × 10
Step 2:= 3 × 55 + 10
Step 3:= 165 + 10

Sum = 175

Common Mistakes & Tips

  • !Forgetting to add the bN term — the constant b contributes b to every term.
  • !Using N(N−1)/2 instead of N(N+1)/2 for Σn from 1 to N.
  • !Confusing the linear coefficient a with the first term a(1) = a + b.

Related Concepts

Arithmetic Sequence

A linear sequence is an arithmetic sequence with first term (a+b) and common difference a.

Sum of Series

General summation of sequence terms.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arithmetic Sequence Calculator

Frequently Asked Questions

How does a linear sequence relate to an arithmetic sequence?

A linear sequence a(n) = an + b is an arithmetic sequence with first term a₁ = a + b and common difference d = a. The two representations are equivalent.

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