Arithmetic Sequence and Series

Arithmetic sequences and series

An arithmetic sequence has a constant difference $d$ between consecutive terms: $a_1, a_1+d, a_1+2d, \ldots$

nth term formula: $a_n = a_1 + (n-1)d$

Sum of first $n$ terms: $S_n = \frac{n(a_1 + a_n)}{2}$ = number of terms × average of first and last.

Step-by-step: Sum $3 + 7 + 11 + 15 + \cdots + 39$

$a_1 = 3$, $d = 4$. First find $n$: $39 = 3 + (n-1)(4)$ → $n = 10$.

$S = \frac{10(3 + 39)}{2} = \frac{10 \times 42}{2} = 210$.

Gauss's trick: Pair first and last: $3 + 39 = 42$, $7 + 35 = 42$, ... 5 pairs × 42 = 210.

Try it in the visualization

Adjust $a_1$, $d$, $n$. The staircase bars show equal steps. Gauss's pairing animates — first+last, second+second-to-last, all summing to the same value. The sum of $n$ terms: $S_n = n(a_1 + a_n)/2$. Gauss's trick: pair first and last terms — each pair sums to $a_1 + a_n$.