Arithmetic progression x Geometric Progression?

Question

Simplify to make a good equation for

S = ∑ (k = 1 to n) { [a_1 + d · (k - 1)] [α_1 · r ^ (k - 1)] }

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a (English letter a) is not identical α (Greek letter Alpha)
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I mean a formula which one can directly substitute without that sigma notation. (Like arithmetic and geometric P both have ther summation formulae)
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Answer 1

That's easy.
S = ∑ (k = 1 to n) { [a_1 + d · (k - 1)] [α_1 · r ^ (k - 1)] } =
a_1 · α_1 · ∑ (k = 1 to n) r ^ (k - 1) +
d ·α_1 · ∑ (k = 1 to n) (k - 1) · r ^ (k - 1) = ...

The first part is ordinary geometric progression. To find the second sum, take the equation for ordinary geometric progression sum and differentiate both parts wrt r. You will get something very similar to this sum on the right.

Drop me a message if this is still unclear.

Answer 2

I feel the equation in the current form is in the most simplified form.

The two expressions AP and GP themselves are in their most simplified forms.If you try multiplying them, you will only end up expanding the equation and there will be no common terms or the like which will help to simplify it. Even if you take a_1 = alpha( i.e the first terms in both the series are equal ), no further simplification seems to be there.