If 5^th and 6^th terms of an A.P. are respectively 6 and 5, find the 11^th term of the A.P.

n^th term of an A.P. is given by,

a_n = a + (n - 1)d

Given,

5^th term = 6

⇒ a₅ = a + (5 - 1)d

⇒ 6 = a + 4d ………(i)

Also,

6^th term = 5

⇒ a₆ = a + (6 - 1)d

⇒ 5 = a + 5d ………(ii)

Subtracting (i) from (ii) we get,

⇒ a + 5d - (a + 4d) = 5 - 6

⇒ a - a + 5d - 4d = -1

⇒ d = -1.

Substituting value of d in (i) we get,

⇒ 6 = a + 4(-1)

⇒ 6 = a - 4

⇒ a = 10.

11^th term of the sequence,

⇒ a₁₁ = 10 + (11 - 1)(-1)

= 10 + 10(-1)

= 10 - 10 = 0.

Hence, 11^th term of the sequence = 0.