4.
In a sequence of positive numbers, the ratio of
each term to the term immediately following it is
1 to 4. What is the ratio of the 3rd term to the 6th
term in this sequence?
(A) 1 to 4
(B) 1 to 16
(C) 1 to 64
(D) 1 to 128
(E) None of the above
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Answers
Answer:
Given
a
n
= 2a
n−1
A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r.
Here, r = 2.
In a G.P.,
a
n
= a
1
r
n−1
,
where,
′
a
n
′
is the n
th
term of the sequence
′
a
1
′
is the 1
st
term of the sequence
′
r
′
is the common ratio
To find the ratio of the 8
th
term to the 5
th
term,
a
8
= a
1
2
8−1
a
8
= 2
7
a
1
a
5
= a
1
2
5−1
a
5
= 2
4
a
1
Ratio of a
8
to a
5
= 2
7
a
1
: 2
4
a
1
=
16
128
: 1
= 8 : 1
Therefore, the ratio of the 8
th
term in the sequence to the 5
th
term is
′
8 to 1
′
.
Step-by-step explanation:
Here, r = 4
To find the ratio of the 3rd term to 6th term is calculated as follows,
a3=a1 4^(3-1)=a1 4^2
a6=a1 4^(6-1)=a1 4^5
we simply the expression as below
a3/a6= a1 4^2/a1 4^5
= (2^2)^2/(2^2)^5
= (2^4)/(2^10)
= 16/1024
= 1/64
there fore,a3:a6=1:64
that is option C
