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Sequences & Series

Arithmetic, geometric, and recursive patterns.

Interactive graph

Interactive Lab

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Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualA pattern where each term increases by a constant difference (d). Like counting by 2s, but starting anywhere.

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

a_n = a_1 + (n-1)d

VisualA pattern where each term is multiplied by a constant ratio (r). Like compound growth.

Algebra $a_n = a_1 \cdot r^{n-1}$ .

a_n = a_1 \cdot r^{n-1}

VisualA series is the sum of the terms of a sequence. Drag the slider for n to watch the sum grow.

AlgebraArithmetic series: $S_n = \frac{n}{2}(a_1 + a_n)$ . Geometric series: $S_n = \frac{a_1(1-r^n)}{1-r}$ .

S_n = \frac{n}{2}(2a_1 + (n-1)d)

Arithmetic nth term

a_n = a_1 + (n-1)d

d is common difference

Geometric nth term

a_n = a_1 r^{n-1}

r is common ratio

Arithmetic Series

S_n = \frac{n}{2}(a_1 + a_n)

Sum of n terms

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easysequences

Find the 10th term of the arithmetic sequence: 2, 5, 8, 11...

easysequences

Find the common ratio of the geometric sequence: 3, 6, 12, 24...