The first four terms of a sequence are shown below:

8, 5, 2, - 1

Which of the following functions best defines this sequence?

f (1) = 8, f (n + 1) = f (n) + 3; for n ≥ 1

f (1) = 8, f (n + 1) = f (n) - 5; for n ≥ 1

f (1) = 8, f (n + 1) = f (n) + 5; for n ≥ 1

f (1) = 8, f (n + 1) = f (n) - 3; for n ≥ 1

Choice D is the correct answer.

Step-by-step explanation:

We have given a arithematic sequence.

8,5,2,-1

We have to find a recurrence relation for given sequence.

The formula for common difference of arithematic sequence is:

f (n+1) - f (n) = d where f (n) and f (n+1) are consecutive terms and d is common difference between consecutive terms.

In given sequence,

f (1) = 8 and f (2) = 5

d = - 3

putting the value of d in above formula, we have

f (n+1) - f (n) = - 3

Adding f (n) to both sides of above equation, we have

f (n+1) - f (n) + f (n) = f (n) - 3

f (n+1) = f (n) - 3

since given that n = 1

f (n+1) = f (n) - 3; for n ≥ 1 which is the answer.

Option 4 is correct.

Step-by-step explanation

Here the first term is 8 so the first term is given by f (1) = 8

Since here the given series is arithmetic series and here common difference is given by 5-8 = - 3

Recurrence relation is given by f (n+1) - f (n) = d

where d is common difference which is given to be - 3.

therefore it is given as

f (n+1) - f (n) = - 3

f (n+1) = f (n) - 3 [ adding f (n) both sides]

Since n is starting from 1 there for its fourth option is correct

f (1) = 8 and, f (n+1) = f (n) - 3), for n ≥1