The value of 'n' for which the nth term of AP's 3, 10, 17 and 63, 65, 67, ... are equal is

The nth term of an Arithmetic Progression (AP) is given by the formula: a + (n-1)d, where 'a' is the first term and 'd' is the common difference.

For the first AP (3, 10, 17, ...), a1 = 3 and d1 = 10 - 3 = 7. So, the nth term of the first AP is: 3 + (n-1)7.

For the second AP (63, 65, 67, ...), a2 = 63 and d2 = 65 - 63 = 2. So, the nth term of the second AP is: 63 + (n-1)2.

We are given that the nth term of both APs are equal. Therefore, we can set the two expressions equal to each other and solve for 'n':

3 + (n-1)7 = 63 + (n-1)2 7n - 4 = 2n + 61 7n - 2n = 61 + 4 5n = 65 n = 65 / 5 n = 13

So, the value of 'n' for which the nth term of the given APs are equal is 13.