Answer

Verified

389.4k+ views

**Hint:**Here, the given term is in geometric progression as the terms are increasing in fixed ratio. So, we will use the concept of Geometric Progression to solve the question. A geometric progression is a sequence or series of numbers where each term after the first is found out by multiplying the previous one by a fixed number called the common ratio.

**Formula used:**

We will use the following formulas:

1. Exponential Formula: \[{a^m} \cdot {a^n} \cdot {a^o} \cdot {a^p}..... = {a^{m + n + o + p + .........}}\]

2. Exponential Formula: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]

3. Geometric Progression is given by \[{S_n} = \dfrac{{a(1 - {r^n})}}{{(1 - r)}}\] , where \[a\] is the first term and \[r\] is the common ratio.

**Complete Step by Step Solution:**

We are given with a geometric Series \[\left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty \].

The given series is of the form \[a \cdot ar \cdot a{r^2} \cdot ....... \cdot a{r^n}\]

Thus, the first term of the Geometric Series \[a = 32\] and \[r = {1^{\dfrac{1}{6}}}\].

By using the formula \[{a^m} \cdot {a^n} \cdot {a^o} \cdot {a^p}..... = {a^{m + n + o + p + .........}}\], we can rewrite the given equation as:

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {32^{\left( {1 + \dfrac{1}{6} + \dfrac{1}{{36}} + .......} \right)}}\]

Now, by applying the formula of Geometric Progression \[{S_n} = \dfrac{{a(1 - {r^n})}}{{(1 - r)}}\] to the power, we get

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {32^{\left( {\dfrac{{1(1 - 0)}}{{1 - \dfrac{1}{6}}}} \right)}}\]

By cross multiplying, we get

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {32^{\left( {\dfrac{1}{{\dfrac{{6 - 1}}{6}}}} \right)}}\]

Simplifying the expression, we get

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {32^{\left( {\dfrac{1}{{\dfrac{5}{6}}}} \right)}}\]

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {32^{\left( {\dfrac{6}{5}} \right)}}\]

Rewriting \[32\] in terms of the power of \[2\], we get

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {2^{^5}}^{\left( {\dfrac{6}{5}} \right)}\]

Now, by using the formula \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], we have

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = {2^6}\]

\[ \Rightarrow \left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty = 64\]

**Therefore, the product \[\left( {32} \right){\left( {32} \right)^{\dfrac{1}{6}}}{\left( {32} \right)^{\dfrac{1}{{36}}}}.......\infty \] is \[64\].**

**Note:**

Here, we need to remember the basics of the Geometric Series and Geometric Sequence. The properties of G.P. are:

1. If every term of G.P. is multiplied or divided by a non-zero number, then the resulting terms are also in G.P.

2. If the common ratio is negative, then the result will alternate between positive and negative.

3. If the common ratio is greater than 1 then there will be an exponential growth towards infinity (positive).

4. If the common ratio is less than \[-1\] then there will be an exponential growth towards infinity (positive and negative).

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE