MQ71 --- Sequence
2013-01-21 5:08 am
MQ71 --- SequenceDifficulty: 40%Given a, b, c are consecutiveterms of an arithmetic sequence, prove thata²(b+ c), b²(c + a), c²(a+ b) are also in arithmetic sequence.
回答 (1)
2013-01-21 5:28 am
✔ 最佳答案
By given , let k = a - b = b - c . a²(b + c) - b²(c + a)
= a²b + a²c - b²c - b²a
= ab(a - b) + c(a² - b²)
= (a - b) (ab + c(a + b))
= (a - b) (ab + bc + ca)
= k (ab + bc + ca)
Similarly ,
b²(c + a) - c²(a + b)
= (b - c) (bc + ca + ab)
= k (ab + bc + ca)
Therefore a²(b+ c), b²(c + a), c²(a+ b) are also in arithmetic sequence.
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