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.
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Here are the solutions to the questions: Question 5: The first term of an Arithmetic Progression is 20. If the fifth term is four-fifth of the third term, find the common difference. Step 1: Write the terms of the Arithmetic Progression (AP). Let the first term be a and the common difference be d. Given a = 20. The third term is a_3 = a + (3-1)d = a + 2d = 20 + 2d. The fifth term is a_5 = a + (5-1)d = a + 4d = 20 + 4d. Step 2: Use the given condition to form an equation. Given that the fifth term is four-fifth of the third term: a_5 = (4)/(5) a_3 20 + 4d = (4)/(5)(20 + 2d) Step 3: Solve the equation for d. Multiply both sides by 5: 5(20 + 4d) = 4(20 + 2d) 100 + 20d = 80 + 8d Subtract 8d from both sides: 100 + 12d = 80 Subtract 100 from both sides: 12d = 80 - 100 12d = -20 d = (-20)/(12) d = -(5)/(3) The correct option is A. The final answer is A. -(5)/(3). Question 6: Evaluate 2 4 + 3 5 - 2. Step 1: Apply the logarithm power rule n a = a^n. 2 4 = 4^ ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._