Answer to Question #313764 in Analytic Geometry for Jane

Question

p,p+7,p+14,.... is arithmetic sequence

Find n term and Find the smallest value of p for which 2021 is a term in the sequence.

Answer 1

$p,p+7,p+14,...$  is arithmetic sequence

$a_1=p, a_2=p+7, a_3=p+14, d=a_2-a_1=7\\ a_n=2021\\ a_n=a_1+d(n-1)\\ 2021=p+7(n-1)\\ 2021=p+7n-7\\ 7n=2028-p\\ n=\frac{2028-p}{7}\\ n\in N\\ n=\frac{2023+5-p}{7}\\ n=289+\frac{5-p}{7}\\ 5-p=7k, k\in Z\\ p=5-7k\\ k=0, p=5\\n=289\\ a_{289}=2021, p=5$