Question

What is the pattern in the sequence $7,15,23,31,39$ ? What are the next two terms?

Answer 1

Step 1: Interpret the given sequence as an $AP$.

The given sequence is $7,15,23,31,39$.

The difference between the first and second terms of the sequence is $15-7=8$

The difference between the second and third terms of the sequence is $23-15=8$

The difference between the third and fourth terms of the sequence is $31-23=8$

The difference between the fourth and fifth terms of the sequence is $39-31=8$

As the difference between every term of the sequence is the same. So, the given sequence is an $AP$ with first term, $a=7$ and common difference, $d=8$

Step 2: Calculate the next two terms of the given sequence

We need to find the sixth and seventh terms of the sequence.

The formula for $n^{th}$ term of an $AP$ is $a_n=a+\left(n-1\right)d$

The sixth term of the $AP$ is $a_6$.

Substitute $n$ as $6$, $a$ as $7$ and $d$ as $8$ in $a_n=a+\left(n-1\right)d$

$$\begin{gathered} a_6=7+\left(6-1\right)8 \\ =7+5\times 8 \\ =7+40 \\ =47 \end{gathered}$$

The seventh term of the AP is $a_7$

Substitute $n$ as $7$, $a$ as $7$ and $d$ as $8$ in $a_n=a+\left(n-1\right)d$

$$\begin{gathered} a_7=7+\left(7-1\right)8 \\ =7+6\times 8 \\ =7+48 \\ =55 \end{gathered}$$

Hence, the next two terms of the given sequence are $47$ and $55$. Also, in each term $8$ is added to the previous term.