The third term of an arithmetic sequence is 14 and the 9th term is - 1. Find the first four terms of the sequence.

The first four terms of the sequence are : 19, 16.5, 14, 11.5

Step-by-step explanation:

In the given sequence:

a (3) = 14, a (9) = - 1

The general term of a sequence in Arithmetic Progression is:

a (n) = a + (n-1) d

a (3) = a + (3 - 1) d = a + 2 d

and a (9) = a + (9 - 1) d = a + 8 d

⇒ a + 2 d = 14 ... (1)

and a + 8 d = - 1 ... (2)

Now, solving the given system of equation, we get:

From (1), a = 14 - 2 d

Put in (2), we get:

a + 8 d = - 1 ⇒ 14 - 2 d + 8d = - 1

⇒ 14 + 6d = - 1

or, 6d = - 1 - 14 = - 15

⇒ d = - 15/6 = - 2.5

or, d = - 2.5

Then a = 14 - 2 d = 14 - 2 (-2.5) = 14 + 5 = 19, or a = 19

Now, first four terms of the sequence is:

a = 19

a (2) = a + 4 = 19 - 2.5 = 16.5

a (3) = a + 2d = 19 + 2 (-2.5) = 19 - 5 = 14

a (4) = a + 3d = 19 + 3 (-2.5) = 19 - 7.5 = 11.5

Hence, the first four terms of the sequence are : 19, 16.5, 14, 11.5