Answer:
n = 8
Step-by-step explanation:
We start by setting up our equation
8748 = 4 * 3^(n-1)
Lets divide by 4:
2187 = 3^(n-1)
To remove 3 and get (n-1) by itself we can take the log_3 of both sides
Log_3 (2187) = n-1
To solve our left side of the equation we can put log(2187) over log(3)
log(2187)/log(3) = n-1 (when we don't have a log base listed its equal to 10)
7 = n - 1
n = 8
Answer:
S₈ = 13120
Step-by-step explanation:
the nth term of a geometric sequence is
[tex]a_{n}[/tex] = a[tex]r^{n-1}[/tex]
a is the first term , r is the common ratio, n the term number
given a = 4, r = 3 and [tex]a_{n}[/tex] = 8748 ( = f(n))
substitute these values into the nth term formula and solve for n
8748 = 4 [tex](3)^{n-1}[/tex] ( divide both sides by 4 )
2187 = [tex]3^{n-1}[/tex]
[ note that 2187 = [tex]3^{7}[/tex] ]
[tex]3^{7}[/tex] = [tex]3^{n-1}[/tex]
Since the bases on both sides are equal, both 3, equate exponents
n - 1 = 7 ( add 1 to both sides )
n = 8
Then the sequence has 8 terms.
The sum to n terms of a geometric sequence is
[tex]S_{n}[/tex] = [tex]\frac{a(r^{n}-1) }{r-1}[/tex] , then
S₈ = [tex]\frac{4(3^{8}-1) }{3-1}[/tex] = [tex]\frac{4(6561-1)}{2}[/tex] = 2(6560) = 13120
