Q:

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

Accepted Solution

A:
Answer:[tex](a) \quad a_1=\sqrt{2};r=2\\(b) \quad a_4=8\sqrt{2}[/tex]Step-by-step explanation:(a) The product of the first two terms is ...   [tex](a_1)(a_2)=a_1(a_1r)=a_1^2r=4[/tex]The product of the first four terms is ...   [tex](a_1)(a_2)(a_3)(a_4)=a_1(a_1r)(a_1r^2)(a_1r^3)=a_1^4r^6=256[/tex]Dividing the second product by the square of the first gives an equation for r.   [tex]\dfrac{a_1^4r^6}{(a_1^2r)^2}=\dfrac{256}{4^2}\\\\r^4=16=2^4\\\\r=2[/tex]Now we can find a1 from the first product.   [tex]a_1^2r=4=2a_1^2\\\\2=a_1^2\\\\a_1=\sqrt{2}[/tex]In summary, ...[tex]a_1=\sqrt{2}\\r=2[/tex]__(b) The fourth term of the sequence is ...   [tex]a_4=a_1r^3=\sqrt{2}\cdot 2^3\\\\a_4=8\sqrt{2}[/tex]