Question

A sequence is defined by the recursive formula f(n + 1) = 1.5f(n). Which sequence could be generated using the formula?

Answer 1

Hello friend,

Geometric sequence -

A geometric sequence is a sequence with the ratio between two consecutive terms constant.

Given-

f(n+1) = 1.5 × f(n)

Solution-

f(n+1) = 1.5 f(n)

f(n+1)/f(n) = 1.5

Here, ratio of two consequtive terms is constant i.e. 1.5

Hence, this is a geometric sequence with common ratio of 1.5.

Hence the sequence will be like,

m, (1.5)m, (1.5^2)m, (1.5^3)m, (1.5^4)m,..... and so on.

Best luck...

Answer 2

Answer:

A geometric sequence with common ratio 1.5.

The n-th term in the sequence is then a × 1.5^(n-1), where a is the first term of the sequence.

One example would be the seqence that starts with the value 1:

1, 1.5, 2.25, 3.375, 5.0625, ...

The n-th term here is 1.5^(n-1).

Step-by-step explanation:

The formula says

f(n+1) =                [ to get a term in the sequence... ]

1.5 f(n)                [ ... multiply the previous term by 1.5 ]

So if the first term is a, then the next is a × 1.5, the next is a × 1.5², the next is a × 1.5³, and so on.

This is precisely what a geometric sequence is.  ( Notice that the ratio between consecutive terms is f(n+1) / f(n) = 1.5, and this is a constant. )

All that's missing for this to fully define a sequence is a starting point.  With different starting point but the same rule, you get different sequences.  But they are all geometric sequences with common ratio 1.5.

For example, starting with 1 we get the sequence:

1, 1.5, 2.25, 3.375, 5.0625, ..., 1.5^(n-1), ...

while starting with 3 gives the sequence

3, 4.5, 6.75, 10.125, 15.1875, ..., 3 × 1.5^(n-1), ...