Question

After the first term in a sequence of positive integers, the ratio of each term to the term immediately preceding it is $2$ to $1$. What is the ratio of the $8^{th}$ term in this sequence to the $5^{th}$ term?

• A. 6 to 1
• B. 8 to 5
• C. 8 to 1
• D. 64 to 1
• E. 256 to 1

Answer 1

The correct option is C 8 to 1

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, ${}^{\prime }$$a_n$${}^{\prime }$ is the $n^{th}$ term of the sequence

${}^{\prime }{a}_1^{\prime }$ is the $1^{st}$ term of the sequence

${}^{\prime }{r}^{\prime }$ 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 $a_5$ $=$ $2^7$ $a_1$ : $2^4$ $a_1$

$=$ $\frac{128}{16}$ : $1$

$=$ $8$ : $1$

Therefore, the ratio of the $8^{th}$ term in the sequence to the $5^{th}$ term is ${}^{\prime }8$ $to$ $1^{\prime }$.