a To write the nth term of the sequence, we need to find the first term and the common ratio.
The first term of the sequence a_n is 2, and the common ratio is 3. Let's write the equation.
a_n= a_1( r)^(n-1) ⇔ a_n= 2( 3)^(n-1)
Let's write nth term of sequence b_n.
The first term of the sequence b_n is 1, and the common ratio is 5. Let's write the equation.
b_n= b_1( r)^(n-1) ⇔ b_n= 1( 5)^(n-1)
We will now check if the sequence formed by the differences of the terms of the sequences a_n and b_n is a geometric sequence.
a_1-b_1, a_2-b_2, a_3-b_3, ...
⇕
2-1, 6-5, 18-25, ...
After subtracting, we get the following.
1, 1, - 7, ...
We see that there is no common ratio. Hence, it is not a geometric sequence.
b Let c_n be the sequence formed by the ratios of the terms of the sequences a_n and b_n.
Since c_1= 2 and r= 35 we can write an equation that represents the nth term of the sequence.
c_n= c_1( r)^(n-1) ⇔ c_n= 2( 3/5)^(n-1)
The common ratio is the ratio of the common ratios of a_n and b_n.
lr_a =3 r_b=5 } ⇒ r_c= 3/5
