c) \( 5 x ; 20 x ; 80 x ; \ldots \) is equal to \( 20480 x \) ?

To determine which term in the sequence \( 5x; 20x; 80x; \ldots \) equals \( 20480x \), we can analyze the sequence as follows: ### Step 1: Identify the Type of Sequence The given sequence is: - First term (\( a_1 \)) = \( 5x \) - Second term (\( a_2 \)) = \( 20x \) - Third term (\( a_3 \)) = \( 80x \) - ... By observing the pattern, each subsequent term is obtained by multiplying the previous term by 4. This is a **geometric sequence** with a common ratio (\( r \)) of 4. ### Step 2: General Formula for the \( n \)-th Term The general formula for the \( n \)-th term of a geometric sequence is: \[ a_n = a_1 \times r^{(n-1)} \] Where: - \( a_n \) = \( n \)-th term - \( a_1 \) = first term - \( r \) = common ratio - \( n \) = term number ### Step 3: Set Up the Equation to Find \( n \) We want to find \( n \) such that: \[ a_n = 20480x \] Substitute the known values into the formula: \[ 20480x = 5x \times 4^{(n-1)} \] ### Step 4: Solve for \( n \) 1. **Divide both sides by \( 5x \):** \[ \frac{20480x}{5x} = 4^{(n-1)} \implies 4096 = 4^{(n-1)} \] 2. **Express 4096 as a power of 4:** \[ 4^6 = 4096 \] Therefore, \[ 4^{(n-1)} = 4^6 \implies n - 1 = 6 \] 3. **Solve for \( n \):** \[ n = 6 + 1 = 7 \] ### Conclusion The term \( 20480x \) is the **7th term** in the sequence. **Answer:** The term 20480x is the seventh term in the sequence.