1. A recursive formula for the sequence 40, 30, 22.5, ... is *

The sequence is an illustration of a geometric sequence

The recursive formula of the sequence is

The sequence is given as: 40, 30, 22.5.....

So, we have:

Rewrite as:

Express 2 as 3 - 1

Substitute n for 3

Hence, the recursive formula is

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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