Question

Find the value of $r$ if $A = 1 + {r^z} + {r^{2z}} + {r^{3z}} + ....\infty $
A. $A{\left( {1 - A} \right)^z}$
B. ${\left( {\dfrac{{A - 1}}{A}} \right)^{\dfrac{1}{z}}}$
C. ${\left( {\dfrac{1}{{A - 1}}} \right)^{\dfrac{1}{z}}}$
D. $A{\left( {1 - A} \right)^{\dfrac{1}{z}}}$

Answer 1

Hint: In this question, we will use the concept of geometric progression to find the value of $r$. First, we will find the common ratio and the first term of the given sequence. Thus, after that we will use the formula of geometric progression to find the value of $r$.

Formula Used: The formula that is used to calculate the sum of infinite terms of geometric progression is $A = \dfrac{a}{{1 - r}}$, Where $a$ is the first term and $r$ is the common ratio.

Complete answer:
 We know that the given series is $A = 1 + {r^z} + {r^{2z}} + {r^{3z}} + ....\infty $
From the above sequence, we can say that
The first term that is $a = 1$
W.K.T common ratio is $\dfrac{r_n}{r_{n-1}}$
So, we can say that the common ratio is $r = {r^z}$
Thus, the sum of infinite terms of geometric progression is $A = \dfrac{a}{{1 - r}}$
By putting the values of first term and common ratio for the given series, we get
$A = \dfrac{1}{{1 - {r^z}}}$
Let us simplify this expression.
Thus, we get
$1 = A\left( {1 - {r^z}} \right)$
$1 = A - A{r^z}$
By simplifying further, we get
$A{r^z} = A - 1$
Now, we will find the value of ${r^z}$ from the above equation.
${r^z} = \dfrac{{A - 1}}{A}$
By taking ${z^{th}}$ root of the above equation, we get
$r = {\left( {\dfrac{{A - 1}}{A}} \right)^{\dfrac{1}{z}}}$

Therefore, the value of $r$ is ${\left( {\dfrac{{A - 1}}{A}} \right)^{\dfrac{1}{z}}}$ if $A = 1 + {r^z} + {r^{2z}} + {r^{3z}} + ....\infty $

Option ‘B’ is correct

Additional Information: A geometric progression is a numerical series of non-zero numbers in which the ratio of any two consecutive terms is always the same. Simply said, the next number in the series is determined by multiplying a given number by the preceding number in the series. Thus, the common ratio is the name given to this constant quantity.

Note: Many times students made a mistake in identifying geometric progression series. The geometric progression is a series in which each subsequent element is produced by multiplying the preceding one by a constant term.