Respuesta :
Question:
A geometric sequence is defined by the equation an = (3)^3 − n. What are the first three terms of the sequence and what is the common ratio, r?
Answer:
The first three terms of sequence is 9, 3 , 1
The common ratio is [tex]\frac{1}{3}[/tex]
Solution:
The geometric sequence is defined by the equation:
[tex]a_n = (3)^{3-n}[/tex]
To find the first three terms of sequence, substitute n = 1, n = 2, n = 3
First term:
Put n = 1 in given equation
[tex]a_1=(3)^{3-1}\\\\a_1 = 3^2\\\\a_1 = 9[/tex]
Thus first term of sequence is 9
Second term:
Put n = 2 in given equation
[tex]a_2 = (3)^{3-2}\\\\a_2 = 3^1\\\\a_2 = 3[/tex]
Thus second term of sequence is 3
Third term:
Put n = 3 in given sequence
[tex]a_3 = (3)^{3-3}\\\\a_3=3^0\\\\a_3 = 1[/tex]
Thus third term of sequence is 1
Thus the first three terms of sequence is 9, 3 , 1
To find common ratio:
Common ratio is found by dividing the two consecutive terms
[tex]a_1 = 9\\\\a_2 = 3\\\\a_3 = 1[/tex]
Thus common ratio is obtained as:
[tex]r = \frac{a_2}{a_1} = \frac{3}{9} = \frac{1}{3}[/tex]
[tex]r = \frac{a_3}{a_2}=\frac{1}{3}[/tex]
Thus common ratio is [tex]\frac{1}{3}[/tex]
