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
Step 1: Define the terms of the arithmetic progression (AP). Let the first term of the AP be a and the common difference be d. Given that the first term of the AP is 2, so a=2. The terms of the AP are: First term: T_1 = a = 2 Second term: T_2 = a+d = 2+d Sixth term: T_6 = a+5d = 2+5d Step 2: Formulate the relationship for the geometric progression (GP). The first, second, and sixth terms of the AP are consecutive terms of a geometric progression. These terms are 2, 2+d, and 2+5d. In a geometric progression, the ratio of consecutive terms is constant. Therefore, we can write: (2+d)/(2) = (2+5d)/(2+d) Step 3: Solve the equation for d. Cross-multiply the terms: (2+d)(2+d) = 2(2+5d) (2+d)^2 = 4+10d Expand the left side: 4 + 4d + d^2 = 4 + 10d Rearrange the terms to form a quadratic equation: d^2 + 4d - 10d + 4 - 4 = 0 d^2 - 6d = 0 Factor out d: d(d-6) = 0 This gives two possible values for d: d=0 or d-6=0 d=6 Step 4: Determine the correct value of d based on the problem statement. The problem states that it is an increasing arithmetic progression. If d=0, the AP terms would be 2, 2, 2, , which is not an increasing progression. If d=6, the AP terms would be 2, 2+6, 2+2(6), , which are 2, 8, 14, . This is an increasing arithmetic progression. Therefore, the common difference must be 6. The common difference of the arithmetic progression is 6.