Mathematics: Arecursive rule for a geometric sequence is a1=8; an=3/4an−1 . what is the explicit rule for this sequence?

A≈314.16cm²Step-by-step explanation:...

Arecursive rule for a geometric sequence is a1=8;

Ответ:

Schoolworkspace453

14.03.2022

Mathematics

The explicit rule for this sequence is; $a_n = 8 \cdot(\frac{3}{4})^{n-1}$

Step-by-step explanation:

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

a_1=8 and $a_n =\frac{3}{4}a_{n-1}$

for n= 2;

$a_2= \frac{3}{4}a_{1}= \frac{3}{4} \cdot 8 = 6$

Similarly for n = 3;

$a_3 = \frac{3}{4}a_{2}= \frac{3}{4} \cdot 6 = \frac{9}{2}$

Therefore, we get a geometric sequence i.e,

$8 , 6 , \frac{9}{2}, .......$

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 , $r = \frac{3}{4}$

then, the explicit rule for this sequence is;

$a_n = 8 \cdot(\frac{3}{4})^{n-1}$

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Arecursive rule for a geometric sequence is a1=8; an=3/4an−1 . what is the explicit rule for this sequence?

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