Question

For what value of n, are the n^th terms of two APs: \(63, 65, 67, …..\)  and  \(3, 10, 17, …..\) equal?

Answer 1

For the first Arithmetic Progression (AP): \(63, 65, 67, ….\) The first term \(a = 63\) and the common difference \(d = a_2 − a_1 = 65 − 63 = 2\). The formula for the n^th term of this AP is: \(a_n = a + (n − 1) d \) Substituting the values: \(a_n = 63 + (n − 1) 2 = 63 + 2n − 2 \) Thus, the n^th term is: \(a_n = 61 + 2n \ \ \ \ ……(1)\) For the second Arithmetic Progression (AP): \(3, 10, 17, ….\) The first term \(a = 3\) and the common difference \(d = a_2 − a_1 = 10 − 3 = 7\). The formula for the n^th term of this AP is: \(a_n = a + (n − 1) d \) Substituting the values: \(a_n = 3 + (n − 1) 7 \) \(a_n = 3 + 7n − 7 \) Thus, the n^th term is: \(a_n = 7n − 4 \ \ \ \ ……(2)\) It is given that the n^th terms of both these APs are equal. Equating equations (1) and (2): \(61 + 2n = 7n − 4\) Rearranging the terms: \(61 + 4 = 7n − 2n\) \(65 = 5n\) Solving for n: \(n = \frac{65}{5}\) \(n = 13\)

Therefore, the 13^th terms of both these Arithmetic Progressions are equal.