Q. 8 - Calculus

Question

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

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

Also, give a closed formula for the sequence.

Step-by-Step Solution

Answer

The first five terms of the sequence are: $\left\{2,4,6,8,10\right\}$.

The closed formula for the sequence is $a_k=2k$ where k is any positive integer.

Step 1. Given Information

A sequence is defined recursively as 

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

Step 2. Find the second and the third term

Substituting $2$ for $k$ in $a_k=a_{k-1}+2$ we get

$$\begin{gathered} a_2=a_{2-1}+2 \\ {a}_2=a_1+2 \\ {a}_2=2+2 \\ {a}_2=4 \end{gathered}$$

Again, substituting $3$ for $k$ in $a_k=a_{k-1}+2$ we get

$$\begin{gathered} a_3=a_{3-1}+2 \\ {a}_3=a_2+2 \\ {a}_3=4+2 \\ {a}_3=6 \end{gathered}$$

Step 3. Find the fourth and the fifth term

In the equation $a_k=a_{k-1}+2$ substitute $4$ for $k$.

$$\begin{gathered} a_4=a_{4-1}+2 \\ {a}_4=a_3+2 \\ {a}_4=6+2 \\ {a}_4=8 \end{gathered}$$

And lastly substitute $5$ for $k$ in the equation $a_k=a_{k-1}+2$ we get

$$\begin{gathered} a_5=a_{5-1}+2 \\ {a}_5=a_4+2 \\ {a}_5=8+2 \\ {a}_5=10 \end{gathered}$$

Step 4. Find the closed formula

Hence the first five terms of the sequence are $\left\{2,4,6,8,10\right\}$

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

Hence, the sequence can be defined by the closed formula $a_k=2k$, where k is the positive integer.