In an arithmetic sequence, a17 = - 40 and

a28 = - 73. Explain how to use this information to write a recursive formula for this sequence

Tn = 2Tn-1 - Tn-2

Step-by-step explanation:

Before we can generate the recursive sequence, we need to find the nth term of the given sequence.

nth term of an AP is given as:

Tn = a + (n-1) d

If a17 = - 40

T17 = a + (17-1) d = - 40

a+16d = - 40 ... (1)

If a28 = - 73

T28 = a + (28-1) d = - 73

a+27d = - 73 ... (2)

Solving both equations simultaneously using elimination method.

Subtracting 1 from 2 we have:

27d - 16d = - 73 - (-40)

11d = - 73+40

11d = - 33

d = - 3

Substituting d = - 3 into 1

a+16 (-3) = - 40

a - 48 = - 40

a = - 40+48

a = 8

Given a = 8, d = - 3, the nth term of the sequence will be

Tn = 8 + (n-1) (-3)

Tn = 8 + (-3n+3)

Tn = 8-3n+3

Tn = 11-3n

Given Tn = 11-3n and d = - 3

Tn-1 = Tn - d ... (3)

Tn-1 = 11-3n + 3

Tn-1 = 14-3n

Tn-2 = Tn-2d ... (4)

Tn-2 = 11-3n-2 (-3)

Tn-2 = 11-3n+6

Tn-2 = 17-3n

From 3, d = Tn - Tn-1

From 4, d = (Tn - Tn-2) / 2

Equating both common difference

(Tn - Tn-2) / 2 = Tn - Tn-1

Tn - Tn-2 = 2 (Tn - Tn-1)

Tn - Tn-2 = 2Tn-2Tn-1

2Tn-Tn = 2Tn-1 - Tn-2

Tn = 2Tn-1 - Tn-2

The recursive formula will be

Tn = 2Tn-1 - Tn-2

The difference between the given terms is

-73 - (-40) = - 33.

The difference between the term numbers is 28 - 17 = 11.

Dividing - 33 / 11 = - 3.

The common difference is - 3.

The recursive formula is the previous term minus 3, or an = an - 1 - 3 where a17 = - 40.