Answer:
Sum = 8.5 + 34.5 Sum = 43 Therefore, the sum of the 4th and 8th terms of the given AP is 43.
Explanation:
AI-generated answerTo find the sum of the 4th and 8th terms of the arithmetic progression (AP), we need to first determine the common difference (d) and then calculate the individual terms. Given: 1. The sum of the 3rd and 7th terms is 30. 2. The sum of the 5th and 9th terms is 56. Step 1: Finding the common difference (d) Let the common difference be d. The sum of the 3rd and 7th terms is 30, so we can write this as an equation: 3rd term + 7th term = 30 Using the formula for the nth term of an AP (an = a1 + (n-1)d), we can write: a1 + 2d + a1 + 6d = 30 Simplifying this equation gives us: 2a1 + 8d = 30 ---(Equation 1) Similarly, the sum of the 5th and 9th terms is 56, so we can write: 5th term + 9th term = 56 Using the formula for the nth term, we can write: a1 + 4d + a1 + 8d = 56 Simplifying this equation gives us: 2a1 + 12d = 56 ---(Equation 2) Step 2: Solving the system of equations Now, we can solve the system of equations formed by Equation 1 and Equation 2 to find the values of a1 and d. Subtracting Equation 1 from Equation 2 gives us: 4d = 26 Dividing both sides of the equation by 2 gives us: 2d = 13 Therefore, the common difference (d) is 13/2 or 6.5. Substituting this value of d into Equation 1, we can solve for a1: 2a1 + 8(6.5) = 30 2a1 + 52 = 30 2a1 = -22 a1 = -11 Step 3: Calculating the 4th and 8th terms Now that we know the common difference (d) and the first term (a1), we can calculate the 4th and 8th terms of the AP. The 4th term (a4) can be found using the formula: a4 = a1 + (4-1)d Substituting the values, we get: a4 = -11 + (3)(6.5) a4 = -11 + 19.5 a4 = 8.5 The 8th term (a8) can be found using the same formula: a8 = a1 + (8-1)d Substituting the values, we get: a8 = -11 + (7)(6.5) a8 = -11 + 45.5 a8 = 34.5 Step 4: Finding the sum of the 4th and 8th terms Finally, we can find the sum of the 4th and 8th terms (a4 + a8):