This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the n^th term of the series 2, 6, 18, 54, 162, we first determine if it is an arithmetic or geometric series. Step 1: Find the common ratio. Divide each term by its preceding term: (6)/(2) = 3 (18)/(6) = 3 (54)/(18) = 3 (162)/(54) = 3 Since the ratio between consecutive terms is constant, this is a geometric series with a common ratio r = 3. Step 2: Identify the first term. The first term of the series is a = 2. Step 3: Use the formula for the n^th term of a geometric series. The formula for the n^th term (T_n) of a geometric series is: T_n = a · r^n-1 Step 4: Substitute the values of a and r into the formula. T_n = 2 · 3^n-1 Step 5: Compare with the given options. The derived n^th term matches option A. The final answer is A.