Sum of Linear Number Sequence Calculator

Calculate the sum of an arithmetic sequence using S = n/2 x (first + last) with step-by-step solutions.

Enter Sequence Parameters

Result

Sum of Sequence
155
First Term (a) 2
Last Term (l) 29
Common Difference (d) 3
Number of Terms (n) 10

Step-by-Step Solution

S = n/2 x (a + l) = 10/2 x (2 + 29) = 155

Understanding Arithmetic Sequences

An arithmetic sequence (or linear number sequence) is a series of numbers in which the difference between consecutive terms is constant. This constant is called the common difference (d). For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.

Arithmetic Sequence Formulas

n-th Term

Find any term in the sequence using the first term and common difference.

a_n = a + (n - 1) x d

Sum (First + Last)

The most common formula when you know the first and last terms.

S = n/2 x (a + l)

Sum (First + Difference)

Use when you know the first term and common difference but not the last term.

S = n/2 x (2a + (n-1)d)

Practical Applications

Arithmetic sequences appear in everyday life: seating arrangements in an auditorium (each row has more seats), salary increments, loan amortization schedules, and stacking patterns. The sum formula attributed to Gauss lets you quickly total large sequences without adding each term individually.

Example

Find the sum of the first 100 natural numbers: a = 1, d = 1, n = 100. Last term l = 100. S = 100/2 x (1 + 100) = 50 x 101 = 5050.