Page 79 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I equAtions

Solution 2: Let a n = ax + by , then a 1 =3,a 2 =7,a 3 = 16,a 4 = 42. Let x, y be the two roots
n
n
2
2
n
of the quadratic equation t − pt − q =0, then x − px − q =0 ⇒ ax n+2 = pax n+1 + qax .
n
n
n
Similarly, by n+2 = pby n+1 + qby . Add them up to obtain ax n+2 + by n+2 = p(ax n+1 + by n+1 )+ q(ax + by ) ⇒ a n+2 = pa n+1 + qa n
n
n
ax n+2 + by n+2 = p(ax n+1 + by n+1 )+ q(ax + by ) ⇒ a n+2 = pa n+1 + qa n .
When n =1, 7p +3q = 16.
When n =2, 16p +7q = 42.
Solve 7p +3q = 16 and 16p +7q = 42 to obtain p = −14,q = 38, thus a n+2 = −14a n+1 + 38a n .
Hence, ax + bx = a 5 = −14 × 42 + 38 × 16 = 20.
5
5
√
Substitute p = −14,q = 38 into the equation x − px − q =0: x + 14x − 38 = 0 ⇒ x = −7 ± 87.
2
2
2
Substitute p = −14,q = 38 into the equation t − pt − q =0: t + 14t − 38 = 0. Since x, y are
2
√
the two roots, then Vieta’s formulas imply x + y = −14, thus y = −14 − x = −7 ∓ 87. Hence,
√ √ √ √
the system has two solutions: (−7+ 87, −7 − 87), (−7 − 87, −7+ 87).

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