Question

The 5th term of an A.P. is thrice the second term, and the 12th term exceeds twice the 6th term by 1. Find the 16th term.

Answer 1

a_n = a + (n − 1)d

The 5^th term (a₅) is thrice the 2^nd term (a₂).

a + 4d = 3(a + d)

a + 4d = 3a + 3d

4d − 3d = 3a − a

d = 2a   ...(1)

The 12^th term (a₁₂) exceeds twice the 6^th term (a₆) by 1.

a + 11d = 2(a + 5d) + 1

a + 11d = 2a + 10d) + 1

11d − 10d = 2a − a + 1

d = a + 1   ...(2)

Since both equations are equal to d, we can set them equal to each other:

2a = a + 1

a = 1

Now, substitute a = 1 into Equation 1.

d = 2a

d = 2 × 1

d = 2

a_n = a + (n − 1)d

a₁₆ = 1 + (16 − 1)2

a₁₆ = 1 + (15)2

a₁₆ = 1 + 30

a₁₆ = 31