Question

In an arithmetic sequence, the sum of the 3rd and 8th term is 1. Given that the sum of the first seven terms is 35, determine the first term and the common difference.

Answer 1

Answer:

           first term:    a₁ = 14

       the common difference:  d = -3

Step-by-step explanation:

The sum of the 3rd and 8th term is:

                                                            $a_3+a_8=1\\\\a_1+2d+a_1+7d=1\\\\\underline{2a_1+9d=1}$

The sum of the first seven terms is 35:

                                                           $S_7=35\\\\\dfrac{a_1+a_7}2\cdot7=35\\{}\qquad\quad^{\div7\qquad\div7}\\ \dfrac{a_1+a_1+d}2=5\\{}\qquad\qquad^{\cdot2\qquad\cdot2}\\2a_1+6d=10\\{}\qquad\ ^{\div2\qquad\div2}\\a_1+3d=5\\{}\quad\ ^{-3d\quad\ -3d}\\a_1=5-3d$

$2a_1+9d=1\\\\2(5-3d)+9d=1\\\\10-6d+9d=1\\{}\qquad\qquad^{-10\quad-10}\\{}\qquad3d=-9\\{}\qquad^{\div3\qquad\div3}\\{}\qquad d=-3\\\\\\a_1=5-3(-3)=5+9=14$