Year 11 Maths - General Number Patterns and Recursion

Using a Recurrence Relation To Generate and Analyse an Arithmetic Sequence

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Questions

Question 1

30638

Given the recurrence relation: \({t_1} = 12,\,\,\,{t_{n + 1}} = \,{t_{n}} + 3\) where \(n \ge 1\), calculate the value of the 20 th term in the sequence.

Answer

\(69\)

Worked Solution

\begin{align}
&\begin{aligned}
t_{1}&=12 \\
t_{2}&=12+3 \\
t_{3}&=12+3+3 \\
&=12+2 \times 3\\
t_{4}&=12+3 \times 3\\
t_{n} &=12+(n-1) \times 3 \\
&=12+3n-3 \\
&=9+3n
\end{aligned}\\
&\begin{aligned}
\therefore t_{20} &=9+3 \times 20 \\
&=69
\end{aligned}
\end{align}

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