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

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

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.