3.2 The first three terms of an arithmetic geometric sequence are 16,8 and 4 respectively. 3.2.1 Determine the \( n^{\text {th }} \) term of the sequence 3.2.2 Determine all possible values of \( n \) for which the sum of the first \( n \) terms of this sequence is greater than 31 3.2.3 Calculate the sum to infinity of this sequence
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The first three terms of the sequence, 16, 8, and 4, suggest a consistent relationship between them. In an arithmetic geometric sequence, each term is produced by multiplying the previous term by a constant (the geometric common ratio) and then adding or subtracting a constant (the arithmetic common difference). Here, the common ratio appears to be \(\frac{1}{2}\), and the common difference is \(-8\). This leads us to determine that the \(n^{\text{th}}\) term can be expressed as: \[ a_n = 16 \cdot \left(\frac{1}{2}\right)^{n-1} - 8(n-1) \] For the sum of the first \(n\) terms, you would generally use the formula for a series sum and then apply your results to the inequality \(S_n > 31\). You would derive the explicit sum formula for your specific sequence and solve it against the conditions set by 31. As for the sum to infinity, considering an arithmetic geometric series does impose some limitations. If the terms head towards zero appropriately, you can utilize the formula for the sum to infinity, taking the convergence of the sequence into account. Specifically, the formula for the sum to infinity \(\frac{a}{1 - r}\) is generally applied when the series converges, providing the geometric part tends to a stable limit. Overall, this combination of arithmetic and geometric properties provides a satisfying way to explore the behavior of the sequence!