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Notes on the maths used in the applet:

An arithmetico-geometric series is a combination of arithmetic and geometric series. It has the general form:

S_n = a + (a + d)r + (a + 2d)r² + … + [a + (n - 1)d] r^n-1

This series can be summed in a similar way to a pure geometric series by multiplying by r and subtracting the result from the original series to obtain:

(1 - r)S_n = a + dr + dr² + … + dr^n-1 - [a + (n - 1)d] rⁿ

Using the expression for the sum of a geometric series, we find:

(1 - r)S_n = a + rd(1 - r^n-1 ) - [a + (n - 1)d] rⁿ (1 - r)

And rearranging:

Sn =	a-[a + (n - 1)d] r (1 - r) + rd(1 - rn-1) (1 – r)2

Convergence:

Infinite arithmetico-geometric series have the same conditions as the geometric series. For |r| < 1, the series converges and for |r| > 1 the series diverges. If the series converges, then the sum of the series is given by;

Sn =	a (1 - r) + rd (1 – r)2