Question

The sum of the three terms of G.P. is \[ \dfrac{{21}}{4} \] and their product is 1 then the common ratio is ……….
A.1
B.2
C.4
D.8

Answer 1

Hint: Given, the sum of three terms and their product. Using this we find the value of \[a \] and then using \[a \] we find the value of \[r \] . We choose the first three terms in G.P. are \[ \dfrac{a}{r} \] , \[a \] and \[ar \] . We choose only the real roots. Here, \[a \] is the first term and \[r \] is the common ratio. G.P. means geometric progression.

Complete step-by-step answer:
Let the first three terms be \[ \dfrac{a}{r} \] , \[a \] and \[ar \] .
Sum of three terms is \[ \dfrac{{21}}{4} \] .
 \[ \Rightarrow \dfrac{a}{r} + a + ar = \dfrac{{21}}{4} \] ------ (1)
Product of three terms is 1
 \[ \Rightarrow \left( { \dfrac{a}{r}} \right)(a)(ar) = 1 \] ------- (2)
Solving (2) we get, \[{a^3} = 1 \] \[ \Rightarrow a = 1 \] (Choosing only real roots).
Now, substitute \[a = 1 \] in equation (1)
 \[ \dfrac{1}{r} + 1 + 1.r = \dfrac{{21}}{4} \]
Multiply by \[r \] on both sides, we get:
 \[ \Rightarrow 1 + r + {r^2} = \dfrac{{21}}{4}r \]
Multiply by 4 on both sides, we get:
 \[ \Rightarrow 4 + 4r + 4{r^2} = 21r \]
Getting all the terms on the left hand side,
 \[ \Rightarrow 4{r^2} + 4r - 21r + 4 = 0 \]
 \[ \Rightarrow 4{r^2} - 17r + 4 = 0 \]
Using factorisation we get,
 \[ \Rightarrow 4{r^2} - 16r - r + 4 = 0 \]
Since 1 is multiple of any number or variable, taking common we get:
 \[ \Rightarrow 4r(r - 4) - 1(r - 4) = 0 \]
Taking common \[(r - 4) \] we get:
 \[ \Rightarrow (4r - 1)(r - 4) = 0 \]
Using zero product property, we get:
 \[ \Rightarrow 4r - 1 = 0 \] or \[r - 4 = 0 \]
 \[ \Rightarrow 4r = 1 \] or \[r = 4 \]
 \[ \Rightarrow r = \dfrac{1}{4} \] or \[r = 4 \]
Therefore, the common ratio is \[r = 4 \] .
So, the correct answer is “r=4”.

Note: We choose the first three terms as mentioned above for any problem. Follow the same procedure as mentioned above. You can also find the three terms using the first term and common ratio. In fact you can find all the terms of geometric progression. While in A.P. we take the first three terms as (a-b) a and (a+b).