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
The formula for the n-th term (last term) of a Geometric Progression (GP) is L = ar^n-1, where a is the first term, r is the common ratio, and n is the number of terms. Step 1: Identify the given values and convert the first term to an improper fraction. First term, a = 5(1)/(3)k = (5 × 3 + 1)/(3)k = (16)/(3)k. Last term, L = (243)/(256)k. Common ratio, r = (3)/(4). Step 2: Substitute these values into the formula L = ar^n-1. (243)/(256)k = (16)/(3)k ((3)/(4))^n-1 Step 3: Simplify the equation by dividing both sides by k (assuming k ≠ 0) and isolating the term with n. (243)/(256) = (16)/(3) ((3)/(4))^n-1 Multiply both sides by (3)/(16): ((3)/(4))^n-1 = (243)/(256) × (3)/(16) ((3)/(4))^n-1 = (243 × 3)/(256 × 16) ((3)/(4))^n-1 = (729)/(4096) Step 4: Express the right side of the equation as a power of (3)/(4). We know that 3^6 = 729 and 4^6 = 4096. So, (729)/(4096) = ((3)/(4))^6. Therefore, the equation becomes: ((3)/(4))^n-1 = ((3)/(4))^6 Step 5: Equate the exponents to solve for n. n-1 = 6 n = 6 + 1 n = 7 The number of terms in the GP is 7. What's next? Send it over!