Question

If we are given $3 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8},a > 0$ then the value of $a$ will be
$A)\dfrac{{15}}{{23}}$
$B)\dfrac{7}{{15}}$
$C)\dfrac{7}{8}$
$D)\dfrac{{15}}{7}$

Answer 1

Hint: First, we need to know about the concept of Arithmetic and Geometric progression.
An arithmetic progression can be given by $a,(a + d),(a + 2d),(a + 3d),...$ where $a$ is the first term and $d$ is a common difference.
A geometric progression can be given by $a,ar,a{r^2},....$where $a$ is the first term and $r$ is a common ratio.
Hence the given question is in the form of geometric progression.
Formula used:
For GP the formula to be calculated $GP = \dfrac{a}{{r - 1}},r \ne 1,r < 0$and $GP = \dfrac{a}{{1 - r}},r \ne 1,r > 0$

Complete step by step answer:
Since from the given equation we have $3 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8},a > 0$
Here the first value $a = 3$and the common ratio is $r = a$
Thus, by applying the sum of the infinite term’s formula $r > 0$, then we get $GP = \dfrac{a}{{1 - r}},r \ne 1,r > 0$
Where $a = 3$and $r = a$, also the GP is given as $3 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8}$
Hence substitute all the know values in above we get $GP = \dfrac{a}{{1 - r}} \Rightarrow 3 + 3a + 3{a^2} + ....\infty = \dfrac{3}{{(1 - a)}}$
Since the value of $3 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8}$, then we get $3 + 3a + 3{a^2} + ....\infty = \dfrac{3}{{(1 - a)}} \Rightarrow \dfrac{{45}}{8} = \dfrac{3}{{(1 - a)}}$
Further solving the equations, we get $\dfrac{{45}}{8} = \dfrac{3}{{(1 - a)}} \Rightarrow 45(1 - a) = 24$
$45(1 - a) = 24 \Rightarrow 45 - 45a = 24$
$45 - 45a = 24 \Rightarrow - 45a = - 21 \Rightarrow a = \dfrac{{ - 21}}{{ - 45}} \Rightarrow \dfrac{7}{{15}}$

So, the correct answer is “Option B”.

Note: Geometric Progression:
In the GP the new series is obtained by multiplying the two consecutive terms so that they have constant factors.
In GP the series is identified with the help of a common ratio between consecutive terms.
Series vary in the exponential form because it increases by multiplying the terms.
The GM is known as the geometric mean which is the mean value or the central term in the set of numbers in the geometric progression. GP of sequence with the n terms is computed as the nth root of the product of all the terms in the sequence taken.
The AM is known as the Arithmetic mean which is the average or mean of the given set of numbers which is computed by adding all the terms in the set of numbers and dividing the sum from the total number of terms in the given.