Question

1. The product of the first two terms of a G.S is 4 and the product of the first four terms in a G.S is 256. (a) Find 1 and r (b) Find 4

Answer 1

`(a) \quad a_1=√(2);r=2\\(b) \quad a_4=8√(2)`

Explanation:

(a) The product of the first two terms is ...

`(a_1)(a_2)=a_1(a_1r)=a_1^2r=4`

The product of the first four terms is ...

`(a_1)(a_2)(a_3)(a_4)=a_1(a_1r)(a_1r^2)(a_1r^3)=a_1^4r^6=256`

Dividing the second product by the square of the first gives an equation for r.

`(a_1^4r^6)/((a_1^2r)^2)=(256)/(4^2)\\\\r^4=16=2^4\\\\r=2`

Now we can find a1 from the first product.

`a_1^2r=4=2a_1^2\\\\2=a_1^2\\\\a_1=√(2)`

In summary, ...

`a_1=√(2)\\r=2`

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(b) The fourth term of the sequence is ...

`a_4=a_1r^3=√(2)\cdot 2^3\\\\a_4=8√(2)`