Question

question 21
for the sequence defined by:
$a_1 = 2$
$a_{n+1} = \frac{5}{a_n} + 2$
find:
$a_2 = $
$a_3 = $
$a_4 = $
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Explanation:

Step1: Calculate \(a_2\)

To find \(a_2\), we use the recursive formula \(a_{n + 1}=\frac{5}{a_n}+2\) with \(n = 1\) (since \(a_1 = 2\)). Substitute \(a_1 = 2\) into the formula:
\(a_2=\frac{5}{a_1}+2=\frac{5}{2}+2\)
First, calculate \(\frac{5}{2}=2.5\), then add 2: \(2.5 + 2 = 4.5=\frac{9}{2}\)

Step2: Calculate \(a_3\)

Now, to find \(a_3\), use the recursive formula with \(n = 2\) (so we use \(a_2=\frac{9}{2}\)). Substitute \(a_2=\frac{9}{2}\) into the formula:
\(a_3=\frac{5}{a_2}+2=\frac{5}{\frac{9}{2}}+2\)
Dividing by a fraction is multiplying by its reciprocal, so \(\frac{5}{\frac{9}{2}}=5\times\frac{2}{9}=\frac{10}{9}\). Then add 2: \(\frac{10}{9}+2=\frac{10}{9}+\frac{18}{9}=\frac{28}{9}\)

Step3: Calculate \(a_4\)

To find \(a_4\), use the recursive formula with \(n = 3\) (using \(a_3=\frac{28}{9}\)). Substitute \(a_3=\frac{28}{9}\) into the formula:
\(a_4=\frac{5}{a_3}+2=\frac{5}{\frac{28}{9}}+2\)
Again, divide by the fraction: \(\frac{5}{\frac{28}{9}}=5\times\frac{9}{28}=\frac{45}{28}\). Then add 2: \(\frac{45}{28}+2=\frac{45}{28}+\frac{56}{28}=\frac{101}{28}\)

Answer:

\(a_2=\frac{9}{2}\) (or \(4.5\)), \(a_3=\frac{28}{9}\) (or approximately \(3.11\)), \(a_4=\frac{101}{28}\) (or approximately \(3.61\))

(If we want to present them as fractions: \(a_2 = \frac{9}{2}\), \(a_3=\frac{28}{9}\), \(a_4=\frac{101}{28}\))