QUESTION 3 In an arithmetic sequence the eighth term first three terms is 12 . Determine the first Evaluate: \( \sum_{t=1}^{12} 3.2^{t-1} \) Determine the value(s) of \( x \) for which the se \( 2+\frac{2}{3}(x+5)+\frac{2}{9}(x+5)^{2}+\ldots \) will converge

In an arithmetic sequence, the first term is denoted as \( a \) and each subsequent term increases by a common difference \( d \). With the eighth term given to be 12, it can be represented as \( a + 7d = 12 \). The first three terms, \( a \), \( a+d \), and \( a+2d \), provide insight into the sequence's progression, allowing you to derive the values of \( a \) and \( d \) based on the relationship between these terms. To evaluate \( \sum_{t=1}^{12} 3.2^{t-1} \), you can use the formula for the sum of a geometric series: \( S_n = a \frac{1 - r^n}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 1 \) (the first term when \( t=1 \)), \( r = 3.2 \), and \( n = 12 \). Plugging in these values, you can calculate the sum accordingly to find the total of this series.