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: