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
Here is the solution for Question 4: Question 4: (i) The first and last terms of an arithmetic progression are 7 and 43 respectively. The sum of the terms of the progression is 250. Step 1: Identify the given values for the arithmetic progression. First term, a = 7 Last term, l = 43 Sum of the terms, S_n = 250 Step 2: Use the formula for the sum of an arithmetic progression to find the number of terms (n). The formula for the sum of an arithmetic progression is: S_n = (n)/(2)(a + l) Substitute the given values into the formula: 250 = (n)/(2)(7 + 43) 250 = (n)/(2)(50) 250 = 25n Divide by 25 to solve for n: n = (250)/(25) n = 10 The number of terms in the progression is 10. Step 3: Use the formula for the n-th term of an arithmetic progression to find the common difference (d). The formula for the n-th term (which is the last term l) is: l = a + (n-1)d Substitute the known values l=43, a=7, and n=10: 43 = 7 + (10-1)d 43 = 7 + 9d Subtract 7 from both sides: 43 - 7 = 9d 36 = 9d Divide by 9 to solve for d: d = (36)/(9) d = 4 The common difference of the progression is 4. The number of terms is 10 and the common difference is 4.