Finding the nth Term of an Arithmetic Sequence

What is an arithmetic sequence?

An arithmetic sequence has a constant difference $d$ between consecutive terms. Each term is obtained by adding $d$ to the previous term: $a_1, a_1+d, a_1+2d, a_1+3d, \ldots$

The formula

The $n$th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Step-by-step solution: Find the 50th term of 3, 7, 11, 15, ...

Step 1 — Identify $a_1$ and $d$.

$a_1 = 3$, and $d = 7 - 3 = 4$.

Step 2 — Apply the formula with $n = 50$:

$a_{50} = 3 + (50 - 1)(4) = 3 + 49 \times 4 = 3 + 196 = 199$

Answer: The 50th term is 199.

Step 3 — Quick check: The 2nd term should be $3 + (2-1)(4) = 7$ ✓. The 4th term: $3 + 3(4) = 15$ ✓.

Why the formula works

Term 1: $a_1$ (add $d$ zero times). Term 2: $a_1 + d$ (add $d$ once). Term 3: $a_1 + 2d$. Term $n$: $a_1 + (n-1)d$ — you add $d$ exactly $(n - 1)$ times to get from term 1 to term $n$.

Common mistakes

• Using $n$ instead of $(n-1)$. $a_{50} = a_1 + 49d$, not $a_1 + 50d$. You add $d$ only 49 times to go from term 1 to term 50.
• Getting $d$ wrong. Always compute $d = a_2 - a_1$ (second term minus first), not vice versa.

Try it in the visualization

Adjust $a_1$, $d$, and $n$. The staircase bar chart shows each term growing by a constant step. The formula is computed live.