+26400 Find a formula for an for the arithmetic sequence a3 = -10, a7 = -18
\(\begin{array}{rcll} \boxed{~ a_n = a_1 + (n-1)d ~ }\\ a_3 &=& a_1 + (3-1)d = -10\\ a_7 &=& a_1 + (7-1)d = -18\\ a_3-a_7 = -10-(-18) &=& a_1 + (3-1)d - [~a_1 + (7-1)d~] \\ -10+18 &=& a_1 + 2d - a_1 - 6d \\ 8 &=& 2d-6d \\ 8 &=& -4d\\ 8 &=& -4d \qquad | \qquad : -4\\ -2 &=& d \\ \mathbf{d} &\mathbf{=}& \mathbf{-2} \\\\ a_3 &=& a_1 + 2d \\ a_1 &=& a_3-2d \qquad a_3=-10 \qquad d = -2\\ a_1 &=& -10-2\cdot (-2) \\ a_1 &=& -10+4 \\ \mathbf{a_1} &\mathbf{=}& \mathbf{-6} \end{array}\)
\(\text{Formula}\\ \boxed{~ \begin{array}{lcll} a_n &=& -6 + (n-1)\cdot(-2) \\ a_n &=& -6 -2n+2\\ a_n &=& -4-2n \end{array} ~}\)
+26400 Find a formula for an for the arithmetic sequence a3 = -10, a7 = -18
\(\begin{array}{rcll} \boxed{~ a_n = a_1 + (n-1)d ~ }\\ a_3 &=& a_1 + (3-1)d = -10\\ a_7 &=& a_1 + (7-1)d = -18\\ a_3-a_7 = -10-(-18) &=& a_1 + (3-1)d - [~a_1 + (7-1)d~] \\ -10+18 &=& a_1 + 2d - a_1 - 6d \\ 8 &=& 2d-6d \\ 8 &=& -4d\\ 8 &=& -4d \qquad | \qquad : -4\\ -2 &=& d \\ \mathbf{d} &\mathbf{=}& \mathbf{-2} \\\\ a_3 &=& a_1 + 2d \\ a_1 &=& a_3-2d \qquad a_3=-10 \qquad d = -2\\ a_1 &=& -10-2\cdot (-2) \\ a_1 &=& -10+4 \\ \mathbf{a_1} &\mathbf{=}& \mathbf{-6} \end{array}\)
\(\text{Formula}\\ \boxed{~ \begin{array}{lcll} a_n &=& -6 + (n-1)\cdot(-2) \\ a_n &=& -6 -2n+2\\ a_n &=& -4-2n \end{array} ~}\)
