Between 3 and 6 insert ten arithmetic means a 1 , … , a 10 , and separately ten harmonic means h 1 , … , h 10 . Statement 1: a 2 h 9 + a 4 h 7 + a 6 h 5 + a 8 h 3 = 72 . Statem

Question

Between 3 and 6 insert ten arithmetic means a 1 ​ , … , a 10 ​ , and separately ten harmonic means h 1 ​ , … , h 10 ​ . Statement 1: a 2 ​ h 9 ​ + a 4 ​ h 7 ​ + a 6 ​ h 5 ​ + a 8 ​ h 3 ​ = 72 . Statement 2: For any two endpoints, the product of the ith arithmetic and ith harmonic mean inserted between them is independent of i.

TestHub · Accepted Answer

Here a i ​ = 3 ( 11 + i ) /11 and h j ​ = 66/ ( 22 − j ) . Hence a i ​ h 11 − i ​ = 18 ; the four specified products total 72. Same-index products are not constant, for example a 1 ​ h 1 ​  = a 2 ​ h 2 ​ .

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