Question

Suppose that a sequence is defined as follows.a₁ = 5, an=−4ªn−1+10 for n>2List the first four terms of the sequence.

Answer 1

The given sequence is defined by:

`a_1=5;a_n=-4a_(n-1)+10\text{ for }n\geqslant2`

Substitute n=2 into the nth term formula:

`\begin{gathered} a_2=-4a_(2-1)+10 \\ \Rightarrow a_2=-4a_1+10 \end{gathered}`

Substitute a₁=5 into the equation:

`a_2=-4(5)+10=-20+10=-10`

Hence, the second term is -10.

Substitute n=3 into the nth term formula:

`\begin{gathered} a_3=-4a_(3-1)+10 \\ \Rightarrow a_3=-4a_2+10 \\ \text{ Substitute }a_2=-10\text{ into the equation:} \\ \Rightarrow a_3=-4(-10)+10=40+10=50 \end{gathered}`

Hence, the third term is 50.

Substitute n=4 into the nth term formula:

`\begin{gathered} a_4=-4a_(4-1)+10 \\ \Rightarrow a_4=-4a_3+10 \\ \text{ Substitute }a_3=50\text{ into the equation:} \\ \Rightarrow a_4=-4(50)+10=-200+10=-190 \end{gathered}`

Hence, the fourth term is -190.

The first four terms of the sequence is: