Question

Hello,Can you please help with question 33 on the photo? Thank you

Answer 1

Explanation

With the help of the given formula, we can find the first four terms of the sequence:

$\begin{gathered} a_1=30 \\ a_2=a_{2-1}-10=a_1-10=20 \\ a_3=a_{3-1}-10=a_2-10=10 \\ a_4=a_{4-1}-10=a_3-10=0 \end{gathered}$

Then, the first four terms of the sequence are 30, 20, 10, 0, ...

Now, as we can see, this is an arithmetic sequence because there is a common difference between each term. The explicit formula of an arithmetic sequence is shown below:

$\begin{gathered} a_n=a_1+d(n-1) \\ \text{ Where} \\ \text{ d is the common difference} \end{gathered}$

Then, we have:

$\begin{gathered} a_1=30 \\ d=-10 \end{gathered}$$\begin{gathered} a_n=a_1+d(n-1) \\ a_n=30-10(n-1) \\ \text{ Apply the distributive property} \\ a_n=30-10*n-10*-1 \\ a_n=30-10n+10 \\ a_n=-10n+40 \end{gathered}$

Thus, a formula for the general term of the sequence is:

$a_{n}=-10n+40$

Now, we substitute n = 20 in the above formula to find the 20th term of the sequence:

$\begin{gathered} a_{n}=-10n+40 \\ a_{20}=-10(20)+40 \\ a_{20}=-200+40 \\ a_{20}=-160 \end{gathered}$Answer

A formula for the general term of the sequence is:

$a_{n}=-10n+40$

The 20th term of the sequence is -160.