+414 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 $a_1$.
+1376
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 $a_1$.
call "d" the amount of addition between successive terms.
starting with a1 then a2 = a1 + d = a1 + d
a3 = a2 + d = (a1 + d) + d = a1 + 2d
a4 = a3 + d = (a1 + d + d) + d = a1 + 3d
a5 = a4 + d = (a1 + d + d + d) + d = a1 + 4d
a6 = a5 + d = (a1 + d + d + d + d) + d = a1 + 5d etc. . . .
given a1 + a3 = 5
a1 + (a1 + 2d) = 5
2a1 + 2d = 5 (eq 1)
given a2 + a4 = 6
(a1 + d) + (a1 + 3d) = 6
2a1 + 4d = 6
a1 + 2d = 3 (eq 2)
Subtract (eq 2) from (eq 1)
2a1 + 2d = 5 (eq 1)
a1 + 2d = 3 (eq 2)
a1 = 2
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