Answer:
The sum of the first nine terms of the sequence is 74.44.
Step-by-step explanation:
Geometric sequence concepts:
The nth term of a geometric sequence is given by the following equation.
[tex]a_{n+1} = ra_{n}[/tex]
In which r is the common ratio.
This can be expanded for the nth term in the following way:
[tex]a_{n} = a_{1}r^{n-1}[/tex]
In which [tex]a_{1}[/tex] is the first term.
Or even:
[tex]a_{n} = a_{m}r^{n-m}[/tex]
The sum of the first n terms of a geometric sequence is given by:
[tex]S_{n} = \frac{a_{1}(1 - r^{n})}{1 - r}[/tex]
Finding the common ratio:
[tex]a_{3} = 3.645, a_{8} = 15[/tex]
[tex]a_{n} = a_{m}r^{n-m}[/tex]
[tex]a_{8} = a_{3}r^{8-3}[/tex]
[tex]a_{3}r^{5} = a_{8}[/tex]
[tex]3.645r^{5} = 15[/tex]
[tex]r^{5} = \frac{15}{3.645}[/tex]
[tex]r = \sqrt[5]{\frac{15}{3.645}}[/tex]
[tex]r = 1.327[/tex]
Finding the first term:
[tex]a_{3} = a_{1}r^{2}[/tex]
[tex]a_{1} = \frac{a_{3}}{r^{2}}[/tex]
[tex]a_{1} = \frac{3.645}{(1.327)^{2}}[/tex]
[tex]a_{1} = 2.07[/tex]
Sum of the first nine terms:
[tex]S_{9} = \frac{2.07*(1 - (1.327)^{9})}{1 - 1.327} = 74.44[/tex]
The sum of the first nine terms of the sequence is 74.44.
Answer:
S₉ = 9.84
Step-by-step explanation:
In a geometric series, the general formula for nth term is:
an = a₁rⁿ⁻¹
where,
an = nth term
a₁ = first term
r = common ratio
From this formula, we have:
a₃ = a₁r² = 3.645 --------- equation (1)
a₈ = a₁r⁷ = 15 ----------- equation (2)
dividing both of these equations, we get:
a₁r⁷/a₁r² = 15/3.645
r⁵ = 4.115
r = (4.115)^1/5
r = 1.33
Now, put this value in equation (1), we get:
a₁(1.33)² = 3.645
a₁ = 0.27
Now, the formula for the sum of geometric series upto nth term is:
Sn = a₁(rⁿ - 1)/(r - 1)
therefore,
S₉ = (0.27)(1.33⁹ - 1)/(1.33 - 1)
S₉ = 9.84
