Solution:
Given the sequence -36, 30, -25, ...
We are required to find the sum of the first 15 terms for the sequence
Firstly, determine what type of sequence it is.
Check if it an arithmetic sequence (AP) or a geometric sequence (GP)
[tex]\begin{gathered} To\text{ test if the sequence is a GP, we have to check if it has a common ratio} \\ r=\frac{T_2}{T_1}=\frac{T_3}{T_2} \\ \\ r=\frac{30}{-36}=\frac{-25}{30} \\ \\ r=-\frac{5}{6}=-\frac{5}{6} \\ The\text{ sequence has a common ratio. Thus, it is a GP} \end{gathered}[/tex]The formula for calculating sum of nth term of a GP is shown below
For this question,
a = -36, r=-5/6, n = 15
The sum of the first 15 terms for the sequence is calculated as follows
[tex]\begin{gathered} S_{15}=\frac{-36(1-(-\frac{5}{6})^{15})}{1-(-\frac{5}{6})} \\ \\ S_{15}=\frac{-36(1.064905)}{1.8333333} \\ \\ S_{15}=-\frac{38.33658}{1.8333333} \\ \\ S_{15}=-20.9108656 \\ S_{15}=-20.91086562 \\ \\ S_{15}=-20.91087\text{ \lparen5 decimal places\rparen} \end{gathered}[/tex]Thus, The sum of the first 15 terms for the sequence -20.911087 (5 decimal places)
