Answer:
First term: 0.5
Sum of the first give terms: 30.5
Step-by-step explanation:
General form of a geometric sequence:
[tex]a_n=ar^{n-1}[/tex]
where:
- [tex]a_n[/tex] is the nth term
- a is the first term
- r is the common ratio
Given terms:
- [tex]a_4=-13.5[/tex]
- [tex]a_5=40.5[/tex]
To find the common ratio r, divide consecutive terms:
[tex]\implies r=\dfrac{a_5}{a_4}=\dfrac{40.5}{-13.5}=-3[/tex]
To find the first term, substitute the found value of r and one of the terms into the general formula:
[tex]\begin{aligned}\implies a_5 =a(-3)^4 & = 40.5\\81a & = 40.5\\a & = \dfrac{40.5}{81}\\a & = 0.5 \end{aligned}[/tex]
Sum of the first n terms of a geometric series:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
To find the sum of the first 5 terms of the geometric sequence, substitute n = 5 and the found values of a and r into the formula:
[tex]\begin{aligned}\implies S_5 & =\dfrac{0.5(1-(-3)^5)}{1-(-3)}\\\\& =\dfrac{0.5(1+243)}{1+3}\\\\& =\dfrac{0.5(244)}{4}\\\\& =\dfrac{122}{4}\\\\ & = 30.5\end{aligned}[/tex]
Therefore, the sum of the first 5 terms is 30.5.
