The 4th term of an A.P is 13 while the 10th term is 31. Find the 21st term

Let's begin by recalling the formula for the nth term of an arithmetic progression (A.P): a_n = a_1 + (n - 1)d where a_n is the nth term of the A.P, a_1 is the first term, n is the number of the term, and d is the common difference between consecutive terms. We are given that the 4th term of the A.P is 13, so we can substitute these values into the formula to get: a_4 = a_1 + (4 - 1)d = 13 Simplifying this equation, we get: a_1 + 3d = 13 ---(1) We are also given that the 10th term of the A.P is 31, so we can use the formula again to get: a_10 = a_1 + (10 - 1)d = 31 Simplifying this equation, we get: a_1 + 9d = 31 ---(2) Now we need to solve for a_1 and d. We can do this by subtracting equation (1) from equation (2) to eliminate a_1: 6d = 18 d = 3 Substituting this value of d into equation (1), we get: a_1 + 3(3) = 13 a_1 = 4 So, the first term of the A.P is 4 and the common difference is 3. Now we can use the formula again to find the 21st term of the A.P: a_21 = a_1 + (21 - 1)d Substituting the values we found earlier, we get: a_21 = 4 + (20)(3) = 64 Therefore, the 21st term of the A.P is 64, and the correct answer is option (C).