Question

If the sequence $\\{ {a_n}\\}$ is in GP, such that $\dfrac{{{a_4}}}{{{a_6}}} = \dfrac{1}{4}$ and ${a_2} + {a_5} = 216$ , then ${a_1}$ is equal to
a.$12$ or $\dfrac{{108}}{7}$
b.$\;10$
c.$7$or $54/7$
d.None of these

Answer 1

In this question, we have been given that the sequence in geometric progression or GP. We know the general form of the geometric expression is $a,ar,a{r^2},a{r^3}...$ , where $a$ is the first term of the progression and $r$ is the common ratio. We should know that we can write ${a_4}$ as ${a_1}{r^3}$ . Similarly, we will write the other term also and then we will first find the common ratio of the given sequence. After this, we will simplify the expression.

Complete answer:
Here we have been given in the question $\dfrac{{{a_4}}}{{{a_6}}} = \dfrac{1}{4}$ .
We can write
${a_4}$ $= {a_1}{r^3}$ .
Similarly, by applying the same method we can write ${a_6} = {a_1}{r^5}$ .
By substituting these in the equation we have:
$\Rightarrow \dfrac{{{a_1}{r^3}}}{{{a_1}{r^5}}} = \dfrac{1}{4}$
We will now simplify this, we can break down the value in the denominator using exponential powers:
$\Rightarrow \dfrac{{{r^3}}}{{{r^2} \times {r^3}}} = \dfrac{1}{4}$
Now we have
$\Rightarrow \dfrac{1}{{{r^2}}} = \dfrac{1}{4}$
By cross multiplying the terms, it gives us
$\Rightarrow {r^2} = 4$
$\Rightarrow r = \sqrt 4$
So we have the value of common ratio i.e.
$r = \pm 2$
It is given that
${a_2} + {a_5} = 216$ .
We will again break down the terms and it can be written as
$\Rightarrow {a_1}r + {a_1}{r^4} = 216$
Let us take case one, where we take the value of $r = 2$
By applying this in the expression, we have:
$\Rightarrow 2{a_1} + {2^4}{a_1} = 216$
$\Rightarrow 2{a_1} + 16{a_1} = 216$
On adding the terms it gives us
$\Rightarrow 18{a_1} = 216$
Therefore we have the value of ${a_1}$ as
$\Rightarrow \dfrac{{216}}{{18}} = 12$
Now let us solve for case two, where we have a negative value of common ratio i.e. $r = - 2$
By applying this in the expression, we have:
$\Rightarrow - 2{a_1} + {( - 2)^4}{a_1} = 216$
$\Rightarrow - 2{a_1} + 16{a_1} = 216$
On simplifying the terms it gives us
$\Rightarrow 14{a_1} = 216$
Therefore we have the value of ${a_1}$ as
$\Rightarrow \dfrac{{216}}{{14}} = \dfrac{{108}}{7}$

Hence the correct option is (a) $12$ or $\dfrac{{108}}{7}$

Note:
We should know that Geometric progression or GP is a type of sequence in which each succeeding term is multiplied by multiplying each preceding term with a mixed number, which is called a common ratio. We should note that the formula of the $nth$ term of a GP is ${a_n} = a{r^{n - 1}}$ .
By applying this we can write
$\Rightarrow {a_4} = a{r^{4 - 1}}$ .
So it can also be written as $a{r^3}$