Question

Find a_1 and d for an arithmetic sequence with these terms. a_4=9and a_7=18

Answer 1

$\bf \begin{array}{llll} term&value\\ -----&-----\\ a_4&9\\ a_5&9+d\\ a_6&(9+d)+d\\ &9+2d\\ a_7&(9+2d)+d\\ &9+3d=\underline{18} \end{array}\\\\ -------------------------------\\\\ 9+3d=18\implies 3d=9\implies d=\cfrac{9}{3}\implies \boxed{d=3}$

$\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ n=7\\ d=3 \end{cases} \\\\\\ a_7=a_1+(7-1)d\implies 18=a_1+(7-1)3\implies 18=a_1+18 \\\\\\ 18-18=a_1\implies \boxed{0=a_1}$