Question

Find the 99th term of the arithmetic sequence 2, -3, -8,…

Answer 1

We have to find the 99th term of the arithmetic sequence: 2,-3, -8...

First, we have to find the explicit function for this arithmetic sequence.

This mean that we have to find the common difference for this sequence:

`\begin{gathered} a_2=a_1+d \\ -3=2_{}+d\Rightarrow d=-3-2=-5 \\ a_3=a_2+d \\ -8=-3+d\Rightarrow d=-8+3=-5 \end{gathered}`

The common difference is d = -5.

We now can find the expression for the explicit function for this sequence as:

`\begin{gathered} a_2=a_1-5 \\ a_3=a_2-5=(a_1-5)-5=a_1+2(-5) \\ a_4=a_3-5=a_1+3(-5) \\ a_n=a_1+(n-1)(-5) \\ a_n=2+(n-1)(-5) \end{gathered}`

Then, we can find the 99th term by calculating this formula for n = 99:

`\begin{gathered} a_(99)=2+(99-1)(-5) \\ a_(99)=2+98(-5) \\ a_(99)=2-490 \\ a_(99)=-488 \end{gathered}`

Answer: the 99th of this sequence is -488.