Mathematics 2000 JAMB Past Questions - Question 13: The 3rd term of an A.P. is 4x-2y and the 9th term is 10x -8y .Find the common difference

Correct Answer

C

Explanation

In an arithmetic progression (A.P.), the general formula for the nth term is given by an​ =a1​+(n−1)dWhere:an​ is the nth term.a1​ is the first term.d is the common difference.n is the term number.You are given that the 3rd term is 4x−2y and the 9th term is 10x−8y.For the 3rd term (n=3), we have:4x−2y=a1​+(3−1)d=a1​+2d    ...(1)For the 9th term (n=9), we have:10x−8y=a1​+(9−1)d=a1​+8d         ....(2)Now, we have a system of two equations (Equations 1 and 2) with two variables (a1​ and d, which is the common difference) that we need to solve simultaneously. Let's do that:Subtract Equation (1) from Equation (2):(10x−8y)−(4x−2y)=(a1​+8d)−(a1​+2d)Simplifying:6x−6y=6dNow, divide both sides by 6 to solve for d:6d=6x−6y​Simplify the right side:d=x−ySo, the common difference of the arithmetic progression is x−y.To find the common difference of an arithmetic progression (A.P.), we can use the formula:nth term = first term + (n - 1) * common differenceLet's use this formula to find the common difference using the given information.The 3rd term of the A.P. is 4x - 2y, which means when n = 3, the nth term is 4x - 2y.4x - 2y = first term + (3 - 1) * common difference4x - 2y = first term + 2 * common differenceSimilarly, the 9th term of the A.P. is 10x - 8y, which means when n = 9, the nth term is 10x - 8y.10x - 8y = first term + (9 - 1) * common difference10x - 8y = first term + 8 * common differenceNow, we have two equations:4x - 2y = first term + 2 * common difference10x - 8y = first term + 8 * common differenceTo find the common difference, we can subtract the first equation from the second equation:(10x - 8y) - (4x - 2y) = (first term + 8 * common difference) - (first term + 2 * common difference)Simplifying the equation:10x - 8y - 4x + 2y = 8 * common difference - 2 * common difference6x - 6y = 6 * common differenceDividing both sides by 6:x - y = common differenceTherefore, the common difference of the arithmetic progression is x - y.