+2653 Let a_1, a_2, a_3, \dots, a_8, a_9, a_{10} be an arithmetic sequence. If $a_1 + a_3 = 5$ and $a_2 + a_4 = 6$, then find $a_1$.
+1217
Let "r" = the amount added to each successive term.
a1 + r = a2 and a2 + r = a3 therefore a3 = a1 + r + r
given that a1 + a3 = 5 therefore (a1) + (a1 + 2r) = 5 (eq 1)
a1 + r = a2 and a2 + r = a3 and a3 + r = a4 therefore a4 = a1 + r + r + r
given that a2 + a4 = 6 therefore (a1 + r) + (a1 + 3r) = 6 (eq 2)
subtract (eq 2) minus (eq 1)
(eq 2) ............................................... 2a1 + 4r = 6
(eq 1) ............................................... 2a1 + 2r = 5
2r = 1
r = 0.5
substitute r back into (eq 2) 2a1 + (4)(0.5) = 6
2a1 = 4
a1 = 2
check answer
a1 = 2.0 , a2 = 2.5 , a3 = 3.0 , a4 = 3.5
a1 + a3 = 5 a2 + a4 = 6
2.0 + 3.0 = 5 2.5 + 3.5 = 6
