Answer:
Step-by-step explanation:
Given the sequence
40, 30, 22.5.......
We want to find the general formula for the sequence.
Let find out if it is arithmetic progression or Geometric progression
Arithmetic progression have a common difference e.g 2,5,8,11...., you will notice they have a common difference of 3.
So, let apply this to the given sequence
40, 30, 22.5.......
d = T2 - T1
d = 30 - 40 = -10
d = T3 - T2
d = 22.5 - 30 = -7.5
So, the difference are not equal, then it is not AP.
Let check for GP, GP have common ratio
r = T2 / T1 = 30 / 40 = 0.75
r = T3 / T2 = 22.5 / 30 = 0.75
Since the common ratio are equal, then, the sequence is GP
The nth term of a G.P is given as
Un = ar^(n-1)
Where
a is first term
r is common ratio
From the given sequence
40, 30, 22.5.......
a = 40 and r = 0.75
Then,
Un = a•r^(n-1)
Un = 40 × 0.75^(n-1)
From indices
a^b / a^c = a^(b-c)
Un = 40 × 0.75ⁿ / 0.75
Un = 53.333 × 0.75ⁿ
Un = 160•0.75ⁿ / 3
That is the nth term at any point
But if we want to find the next term given we know the previous term
From G.P
U1 = a
U2 = U1 × r = ar
U3 = U2 × r = ar × r = ar²
So, in this case r = 0.75 = ¾
So, the previous term multiply by ¾ will give the next term
Then,
Un = U(n-1) × 0.75
Un = ¾ U(n-1)
Un is next term
U(n-1) is previous term
The sequence is an illustration of a geometric sequence
The recursive formula of the sequence is [tex]\mathbf{a_n = 40 \times 0.75^{n-1}}[/tex]
The sequence is given as: 40, 30, 22.5.....
So, we have:
[tex]\mathbf{a_1 = 40}[/tex]
[tex]\mathbf{a_2 = 30}[/tex]
[tex]\mathbf{a_3 = 22.5}[/tex]
Rewrite as:
[tex]\mathbf{a_1 = 40}[/tex]
[tex]\mathbf{a_2 = 40 \times 0.75}[/tex]
[tex]\mathbf{a_3 = 40 \times 0.75^2}[/tex]
Express 2 as 3 - 1
[tex]\mathbf{a_3 = 40 \times 0.75^{3-1}}[/tex]
Substitute n for 3
[tex]\mathbf{a_n = 40 \times 0.75^{n-1}}[/tex]
Hence, the recursive formula is [tex]\mathbf{a_n = 40 \times 0.75^{n-1}}[/tex]
Read more about recursive formula at:
https://brainly.com/question/11679190
