Arithmetic Sequences

An arithmetic sequence adds a fixed amount, the common difference d, each step.

4, 7, 10, 13, … has d = 3.

To find any term without listing them all:
aₙ = a₁ + (n − 1)d

The 20th term of that sequence: a₂₀ = 4 + 19(3) = 61.

The (n − 1) trips people up. The first term needs zero jumps, the second needs one, so the nth needs n − 1.

Find d by subtracting any term from the one after it. If those differences are not all equal, it is not arithmetic.

An arithmetic sequence satisfies the recursive definition aₙ = aₙ₋₁ + d with a₁ given, and the explicit formula aₙ = a₁ + (n − 1)d.

The explicit form is a linear function restricted to the positive integers: rewriting it as aₙ = dn + (a₁ − d) shows slope d and intercept a₁ − d. That is why plotted terms are collinear and why arithmetic sequences and linear functions share their structure — the difference is a discrete domain rather than a continuous one.

The sum of the first n terms is Sₙ = n(a₁ + aₙ)/2, the count times the average of the first and last terms. Gauss's pairing argument gives it directly: adding the sequence to its own reversal pairs every position into the same total a₁ + aₙ, and there are n such pairs counted twice.

Both formulas can be solved for any variable, so problems asking "which term equals 61" are handled by substituting and solving for n — with a non-integer result meaning the value never appears.