# The sum of 3 consecutive terms of an arithmetic series is 15, the product of the same 3 terms is 45. Can the series be determined.

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### 1 Answer

The sum of three consecutive terms of a series are a, a + d and a + 2d.

Their sum is 15, a + a + d + a + 2d = 15

=> 3a + 3d = 15

=> a + d = 5

=> a = 5 - d

The product of the same three terms is 45

a*(a + d)*(a + 2d) = 45

=> (5 - d)*5*(5 - d + 2d) = 45

=> 5*(5 - d)(5 + d) = 45

=> 5*(25 - d^2) = 45

=> 125 - 5d^2 = 45.

=>5d^2 = 80

=> d^2 = 16

=> d = 4 or d = -4

With d = 4, a = 1

The terms of the series have a common difference of 4 and one of the terms is 1.

With d = -4, a = 9

The terms of the series have a common difference of -4 and one of the terms is 9

**Two arithmetic series satisfy the given conditions, one has a common difference of 4 and one of the terms is 1 and the other has a common difference of -4 and one of the terms is 9.**