\#1. The first five terms of a pattern are shown. \( 10,20,40,80 \) The equation \( S_{n}=a(b)^{n} \) can be used to calculate the value, \( S_{n} \), of any term, \( n \), in the pattern. NOTE - this is EXPONENTIAL (not a geometric sequence formula) What is the value of \( a \) in the equation? What is the value of \( b \) in the equation? What is the value of \( S_{6} \) ?

Below is a step-by-step solution: --- **Step 1. Write the given pattern and formula** We are given the first few terms of the pattern: \[ 10,\quad 20,\quad 40,\quad 80, \dots \] and the exponential formula: \[ S_{n} = a(b)^{n} \] --- **Step 2. Identify the value of \( b \)** Since each term is obtained by multiplying the previous term by a constant, we compare the ratio of consecutive terms. For example, \[ \frac{S_{2}}{S_{1}} = \frac{20}{10} = 2 \] Thus, \[ b = 2. \] --- **Step 3. Determine the value of \( a \)** Using the first term, where \( S_{1} = 10 \), we substitute \( n = 1 \) into the formula: \[ S_{1} = a(b)^{1} = a \cdot b. \] Substitute the known values: \[ 10 = a \cdot 2. \] Solve for \( a \): \[ a = \frac{10}{2} = 5. \] --- **Step 4. Calculate \( S_{6} \)** Now that we have found \( a = 5 \) and \( b = 2 \), we substitute these into the formula to find the sixth term: \[ S_{6} = 5 \cdot 2^{6}. \] Since \[ 2^6 = 64, \] we have \[ S_{6} = 5 \cdot 64 = 320. \] --- **Final Answers:** - The value of \( a \) is \( \boxed{5} \). - The value of \( b \) is \( \boxed{2} \). - The value of \( S_{6} \) is \( \boxed{320} \).