Finding the next term in a sequence.

Let $d$ denote the common difference of the arithmetic sequence and $r$ denote the common ratio of the geometric sequence. Then: \begin{align*} c_1 &= a_1 - b_1\\ c_2 &= a_2 - b_2\\ &= (a_1 + d) - (b_1r)\\ c_3 &= a_3 - b_3\\ &= (a_1 + 2d) - (b_1r^2)\\ c_4 &= a_4 - b_4\\ &= (a_1 + 3d) - (b_1r^3) \end{align*} And you want to find this value: $c_5 = a_5-b_5 = (a_1+4d) - (b_1r^4)$.

We have this system: \begin{align*} 2 &= a_1 - b_1\\ 8 &= a_1 + d - b_1r\\ 6 &= a_1 + 2d - b_1r^2\\ 20 &= a_1 + 3d - b_1r^3 \end{align*} It's mildly unpleasant but it's four equations with four variables, and there is indeed a unique solution. Can you take it from here?

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Sequences And Series

Algebra Precalculus

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