Question

A sequence is defined recursively by the given formulas. Find the first five terms of the sequence.

$a_n=2(a_{n-1}+2)and{a}_1=3$

Answer 1

Find first five terms of the series:

$a_n=2(a_{n-1}+2)and{a}_1=3$,

To find $1st$ five terms of series

$a_1,a_2,a_3,a_4,a_5$,

$$\begin{gathered} for{a}_2 \\ \Rightarrow a_2=2(a_1+2) \\ \Rightarrow a_2=2(3+2) \\ \Rightarrow a_2=10 \end{gathered}$$…1

$$\begin{gathered} for{a}_3 \\ \Rightarrow a_3=2(a_2+2) \\ \Rightarrow a_3=2(10+2) \\ \Rightarrow a_3=24 \end{gathered}$$…2

$$\begin{gathered} for{a}_4 \\ \Rightarrow a_4=2(a_3+2) \\ \Rightarrow a_4=2(24+2) \\ \Rightarrow a_4=42 \end{gathered}$$…..3

$$\begin{gathered} for{a}_5 \\ \Rightarrow a_5=2(a_4+2) \\ \Rightarrow a_5=2(42+2) \\ \Rightarrow a_5=88 \end{gathered}$$……4

Form $1,2,3,4$

Hence ,we get series of $1st$ five values,

$3,10,24,42,88$.