3.) The sum of the first 8 terms of an AP is 68, and the sum of the first 16 terms is 264. What is the common difference? a) 2 b) 4 SOLUTION: c) 6 d) 8
Answer (1)
Answer:To find the common difference of an arithmetic progression (AP) given the sums of its first 8 and 16 terms, we can use the following approach.Let \(a\) be the first term and \(d\) be the common difference of the AP.The formula for the sum of the first \(n\) terms of an AP is:\[ S_n = \frac{n}{2} \left[ 2a + (n-1)d \right] \]Given:1. The sum of the first 8 terms is 68:\[ S_8 = \frac{8}{2} \left[ 2a + (8-1)d \right] = 68 \]\[ 4 \left[ 2a + 7d \right] = 68 \]\[ 2a + 7d = 17 \quad \text{(Equation 1)} \]2. The sum of the first 16 terms is 264:\[ S_{16} = \frac{16}{2} \left[ 2a + (16-1)d \right] = 264 \]\[ 8 \left[ 2a + 15d \right] = 264 \]\[ 2a + 15d = 33 \quad \text{(Equation 2)} \]Now, solve these two equations:**Subtract Equation 1 from Equation 2:**\[ (2a + 15d) - (2a + 7d) = 33 - 17 \]\[ 8d = 16 \]\[ d = 2 \]Thus, the common difference \(d\) is \(2\). So the correct answer is **a) 2**.Step-by-step explanation:kindly brainly me, thank you. happy learning! :)
