Question

Three terms of an arithmetic sequence are shown below. Which recursive formula defines the sequence? f(1) = 6, f(4) = 12, f(7) = 18 f (n + 1) = f(n) + 6 f (n + 1) = 2f(n) f (n + 1) = f(n) + 2 f (n + 1) = 1.5f(n)

Answer 1

Given that f(1) = 6,  f(4) = 12, f(7) = 18

nth term in the series is define as:

an = a1 + (n - 1)d

Given that the first term, a1,  is 6:

an =6 + (n - 1)d

Given that the 4th term, a4, is 12:

12 = 6 + 3d

3d = 12 - 6

3d = 6

d = 2

Find nth term:

an =6 + (n - 1)2

an = 6 + 2n - 2

an = 4 + 2n

Write in function form:

f(n) = 4 + 2n

Find n + 1 term:

f(n + 1) = 4 + 2(n + 1)

Distribute 2:

f(n + 1) = 4 + 2n + 2

Sub f(n) = 4 + 2n:

f(n + 1) = f(n) + 2

Answer: f(n + 1) = f(n) + 2

Answer 2

Given,
f(1) = 6 , f(4) = 12 and f(7) = 18

options are given,
1. f(n + 1) = f(n) + 6
2. f(n + 1) = 2f(n)
3. f(n + 1) = f(n) + 2
4. f(n + 1) = 1.5f(n)

Let an Arithmetic sequence is in the form of $f(n)=a+(n-1)d$

now, f(1) = a + (1 - 1)d = 6
a = 6

so, f(n) = 6 + (n - 1)d

again take n = 4,
f(4) = 6 + (4 - 1)d = 12 [ as given f(4) = 12]
6 + 3d = 12
3d = 6 => d = 2

so we get, f(n) = 6 + (n - 1)2 = 6 + 2n - 2
f(n) = 4 + 2n
you can check, this expression is true or false by taking n = 7
f(7) = 4 + 2 × 7 = 18 , hence it is true.

now, f(n) = 4 + 2n
f(n + 1) = 4 + 2(n + 1) = 4 + 2n + 2
f(n + 1) = (4 + 2n) + 2 = f(n) + 2
hence, f(n + 1) = f(n) + 2
option (2) is correct.