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Question: If we are given \(3 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8},a > 0\) then the value of \(a\) wil...

If we are given 3+3a+3a2+....∞=458,a>03 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8},a > 0 then the value of aa will be
A)1523A)\dfrac{{15}}{{23}}
B)715B)\dfrac{7}{{15}}
C)78C)\dfrac{7}{8}
D)157D)\dfrac{{15}}{7}

Explanation

Solution

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),...a,(a + d),(a + 2d),(a + 3d),... where aa is the first term and dd is a common difference.
A geometric progression can be given by a,ar,ar2,....a,ar,a{r^2},....where aa is the first term and rr 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=ar−1,r≠1,r<0GP = \dfrac{a}{{r - 1}},r \ne 1,r < 0and GP=a1−r,r≠1,r>0GP = \dfrac{a}{{1 - r}},r \ne 1,r > 0

Complete step by step answer:
Since from the given equation we have 3+3a+3a2+....∞=458,a>03 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8},a > 0
Here the first value a=3a = 3and the common ratio is r=ar = a
Thus, by applying the sum of the infinite term’s formula r>0r > 0, then we get GP=a1−r,r≠1,r>0GP = \dfrac{a}{{1 - r}},r \ne 1,r > 0
Where a=3a = 3and r=ar = a, also the GP is given as 3+3a+3a2+....∞=4583 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8}
Hence substitute all the know values in above we get GP=a1−r⇒3+3a+3a2+....∞=3(1−a)GP = \dfrac{a}{{1 - r}} \Rightarrow 3 + 3a + 3{a^2} + ....\infty = \dfrac{3}{{(1 - a)}}
Since the value of 3+3a+3a2+....∞=4583 + 3a + 3{a^2} + ....\infty = \dfrac{{45}}{8}, then we get 3+3a+3a2+....∞=3(1−a)⇒458=3(1−a)3 + 3a + 3{a^2} + ....\infty = \dfrac{3}{{(1 - a)}} \Rightarrow \dfrac{{45}}{8} = \dfrac{3}{{(1 - a)}}
Further solving the equations, we get 458=3(1−a)⇒45(1−a)=24\dfrac{{45}}{8} = \dfrac{3}{{(1 - a)}} \Rightarrow 45(1 - a) = 24
45(1−a)=24⇒45−45a=2445(1 - a) = 24 \Rightarrow 45 - 45a = 24
45−45a=24⇒−45a=−21⇒a=−21−45⇒71545 - 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.