The sixth term of an A.P. is 12 and its eighth term is 22. Locate its first…

Given: 6th term = 12 and 8th term = 22.

Formula for n-th term of an A.P.: Tn = a + (n − 1)d

1. Write the two equations from the given terms:

   T6 = a + 5d = 12

   T8 = a + 7d = 22

2. Subtract the first equation from the second to eliminate a:

   (a + 7d) − (a + 5d) = 22 − 12 ⇒ 2d = 10 ⇒ d = 5

3. Substitute d back into one equation to find a:

   a + 5d = 12 ⇒ a + 25 = 12 ⇒ a = −13

4. Find the 16th term:

   T16 = a + 15d = −13 + 15·5 = −13 + 75 = 62

Final answers: first term = −13, common difference = 5, 16th term = 62