Question

Find an arithmetic progression of length six beginning

with 7 that contains only primes.

Short answer

The arithmetic progression beginning with 7 , which only contains prime is 7,37,67,97,127,157

Step-by-step solution

Prime number.

prime: an integer greater than 1 with exactly two positive integer divisors 1 and the number itself.

Finding arithmetic progression

We know an arithmetic progression is of the form

$a_n=a+(n-1)d$

Since we need to find arithmetic progression starting with 7 therefore

a = 7

Now we have to find prime values of $a_n$,

Therefore, taking d = 30 n= 2 ,

substituting the values

$\begin{array}{r}{a}_2=7+(2-1)\left(30\right)\\ =37\end{array}$

For, n = 3

$\begin{array}{r}{a}_3=7+(3-1)\left(30\right)\\ =67\end{array}$

For, n = 4

$\begin{array}{r}{a}_3=7+(4-1)\left(30\right)\\ =97\end{array}$

For, n = 5

$\begin{array}{r}{\mathrm{a}}_3=7+(5-1)\left(30\right)\\ =127\end{array}$

For, n = 6

$\begin{array}{r}{a}_3=7+(6-1)\left(30\right)\\ =157\end{array}$

Hence the arithmetic progression beginning with 7 which only contains prime is 7,37,67,97,127,157