Geometric SequenceFind the terms a1 and a3 for the following following geometric sequence:  a1, -3, a3, -4/3 , ..

Expert Answers
sciencesolve | Certified Educator

You should use the property of geometric series, such that:

`{(a_2 = a_1*r),(a_4 = a_3*r):}`

You need to substitute `-3` for `a_2` and `-4/3` for `a_4` such that:

`{(-3 = a_1*r),(-4/3 = a_3*r):}`

You may also use `a_3 = a_2*r` , such that:

`a_3 = -3*r => -4/3 = (-3*r)*r => r^2 = 4/9 => r_(1,2) = +-2/3`

`r = 2/3 => a_1 = (-3/(2/3)) => a_1 = -9/2`

`r = 2/3 => a_3 = -3*2/3 => a_3 = -2`

`r = -2/3 => a_1 = (-3/(-2/3)) => a_1 = 9/2`

`r = -2/3 => a_3 = -3*(-2/3) => a_3 = 2`

Hence, evaluating the terms `a_1` and` a_3` yields` a_1 = +-9/2` and `a_3 = +-2.`

giorgiana1976 | Student

We'll use the theorem of geometric mean of the terms of a geometric series.

(-3)^2 = a1*a3

We'll use the symmetric property and we'll get:

a1*a3 = 9

We also could write:

a3^2 = (-3)*(-4/3)

We'll simplify and we'll get:

a3^2 = 4

a3 = sqrt 4

a3 = 2 or a3 = -2

For a3 = 2, we'll get:

a1*2 = 9

a1 = 9/2

For a3 = -2, we'll get:

a1 = -9/2

The geometric series are: {-9/2 , -3 , -2 , -4/3} or {9/2 , -3 , 2 , -4/3}.

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