Question

5. Find the sum of the first 15 terms for the sequence (-36, 30, -25,... } to 5 decimal places past the decimal point.-20.910870-174,454.6652-201.98042

Answer 1

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)

`\begin{gathered} To\text{ test if the sequence is a GP, we have to check if it has a common ratio} \\ r=(T_2)/(T_1)=(T_3)/(T_2) \\ \\ r=(30)/(-36)=(-25)/(30) \\ \\ r=-(5)/(6)=-(5)/(6) \\ The\text{ sequence has a common ratio. Thus, it is a GP} \end{gathered}`

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

`\begin{gathered} S_(15)=(-36(1-(-(5)/(6))^(15)))/(1-(-(5)/(6))) \\ \\ S_(15)=(-36(1.064905))/(1.8333333) \\ \\ S_(15)=-(38.33658)/(1.8333333) \\ \\ S_(15)=-20.9108656 \\ S_(15)=-20.91086562 \\ \\ S_(15)=-20.91087\text{ \lparen5 decimal places\rparen} \end{gathered}`

Thus, The sum of the first 15 terms for the sequence -20.911087 (5 decimal places)