[tex]a_n = -4 \times 6^{n-1}[/tex]
Solution:
Given that,
-4, -24, -144, -864
To find: recursive formula for geometric sequence
Find the common ratio "r"
[tex]r = \frac{-24}{-4} = 6\\\\ r = \frac{-144}{-24} = 6[/tex]
The nth term of geometric sequence is given as:
[tex]a_n = a_1 \times r^{n-1}[/tex]
Where,
n is the nth term
[tex]a_1[/tex] is the first term
r is the common ratio
From sequence,
[tex]a_1 = -4\\\\r = 6[/tex]
Therefore,
[tex]a_n = -4 \times 6^{n-1}[/tex]
Where, n = 1, 2, 3, ....
Thus the recursive formula is found
The recursive formula is [tex]a_n=6(a_{n-1})[/tex]
Explanation:
The sequence is [tex]-4,-24,-144,-864, \dots[/tex]
Since, it is given that it is a geometric sequence, let us find the common ratio of the sequence.
Thus, we have,
[tex]r=\frac{-24}{-4} =6[/tex]
Also,
[tex]r=\frac{-144}{-24} =6[/tex]
And,
[tex]r=\frac{-864}{-144} =6[/tex]
Hence, dividing each term of the sequence, the common ratio is [tex]r=2[/tex]
Now, we shall determine the recursive formula for this geometric sequence.
The formula to find the recursive formula for the geometric sequence is given by
[tex]a_n=r(a_{n-1})[/tex]
Substituting the value of r, we get,
[tex]a_n=6(a_{n-1})[/tex]
Therefore, the recursive formula is [tex]a_n=6(a_{n-1})[/tex]
