Q. 7 - Calculus

Question

Give the first five terms of the following recursively defined sequence:

 $a_1=1$, and $a_k=a_{k-1}+2$ for $k\ge 2$.

Also, give a closed formula for the sequence. 

Step-by-Step Solution

Answer

$$\begin{gathered} \{1,3,5,7,9\} \\ {a}_k=2k-1; k\ge 1 \end{gathered}$$

Step1. Given Information

$\text{ Consider the sequence }{a}_k=a_{k-1}+2,a_1=1,k\ge 2$

$\text{ The objective is to find the first five terms of the sequence. Also, find the closed formula. }$

$\text{ To find the terms of the sequence, substitute }k=2,3,4,5\text{ in }{a}_k=a_{k-1}+2$

Step2. Substitution

$a_2=a_{2-1}+2$

$=a_1+2$ (using $a=1$)

$=1+2$

Therefore, $a_2=3$

Step3. Substitution

Substituting $k=3$,

$a_3=a_{3-1}+2$

$=a_2+2$  using($a_2=3$)

$=3+2$

$=5$

Therefore, $a_3=5$

Step4. Substitution

Substituting $k=4$,

$a_4=a_{4-1}+2$

$$\begin{gathered} =a_3+2 u\sin g(a_3=5) \\ =5+2 \\ =7 \end{gathered}$$

Therefore, $a_4=7$

Step5. Substitution

Substituting $k=5$

$$\begin{gathered} a_5=a_{5-1}+2 \\ =a_4+2 u\sin g(a_4=7) \\ =7+2 \\ =9 \end{gathered}$$

Therefore, $a_5=9$

Step6. Answer

Hence, the five terms of the sequence are $\{1,3,5,7,9\}$.

Clearly, the given sequence is sequence of consecutive odd numbers. 

$\text{ Hence, the closed formula is }{a}_k=2k-1\text{, where }k\text{ is the positive integer and }k\ge 1\text{. }$