Question

A recursive rule for a geometric sequence is a1=8;an=3/4an−1 .

What is the explicit rule for this sequence?

Answer 1

8(3/4)^n-1 i hope this help....

Answer 2

The explicit rule for this sequence is;
`a_n = 8 \cdot((3)/(4))^(n-1)`

Explanation:

Given the statement: A recursive rule for a geometric sequence is

`a_1=8` and
`a_n =(3)/(4)a_(n-1)`

for n= 2;

`a_2= (3)/(4)a_(1)= (3)/(4) \cdot 8 = 6`

Similarly for n = 3;

`a_3 = (3)/(4)a_(2)= (3)/(4) \cdot 6 = (9)/(2)`

Therefore, we get a geometric sequence i.e,

`8 , 6 , (9)/(2), .......`

Now, to find the explicit rule for this geometric sequence:

A geometric sequence states that a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio(r).

It is given by:
`a_n = a_1r^(n-1)` where
`a_1` is the first term, r s the common ratio and n is the number of terms;

In the given sequence:
`a_1 = 8`,
`r = (3)/(4)`

then, the explicit rule for this sequence is;

`a_n = 8 \cdot((3)/(4))^(n-1)`