Question

A sequence is defined by the term-to-term rule

Un+1 = U₂ - 3u + 1
Given that u₁ = 3,
a) find u₂
b) find us
c) find u₁

Answer 1

a) u₂ = 1

b) u₃ = -1

c) u₄ = 5

Explanation:

A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Given recursive rule:

`\begin{cases}u_(n+1)=u{_n}^2-3u_n+1\\u_1=3\end{cases}`

Part a

To find u₂, substitute n = 1 into the formula:

`\begin{aligned}\implies u_(2)=u_(1+1)&=u{_1}^2-3u_1+1\\&=(3)^2-3(3)+1\\&=9-9+1\\&=0+1\\&=1\end{aligned}`

Part b

To find u₃, substitute n = 2 into the formula:

`\begin{aligned}\implies u_(3)=u_(2+1)&=u{_2}^2-3u_2+1\\&=(1)^2-3(1)+1\\&=1-3+1\\&=-2+1\\&=-1\end{aligned}`

Part c

To find u₄, substitute n = 3 into the formula:

`\begin{aligned}\implies u_(4)=u_(3+1)&=u{_3}^2-3u_3+1\\&=(-1)^2-3(-1)+1\\&=1+3+1\\&=4+1\\&=5\end{aligned}`