Answer
Verified
394.5k+ views
Hint: Let $S = \dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....$. Simplify the equation by adding 1 on both sides. Use the expansion \[{(1 - x)^{ - \dfrac{p}{q}}} = 1 + \dfrac{p}{{1!}}(\dfrac{x}{q}) + \dfrac{{p(p + q)}}{{2!}}{(\dfrac{x}{q})^2} + ....\] and compare its RHS with that of S to get the value of x and hence find the value of S.
Complete step by step solution:
We have an infinite series $\dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....$
We need to find the sum of this series.
Let’s call the sum as S.
Then we have $S = \dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....(1)$
We will simplify the above expression by adding 1 on both the sides of the equation.
Thus, we have
$1 + S = 1 + \dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....$
Now, RHS can be expressed as follows:
$1 + S = 1 + \dfrac{3}{{1!}}.(\dfrac{1}{4}) + \dfrac{{3.5}}{{2!}}.{(\dfrac{1}{4})^2} + \dfrac{{3.5.7}}{{3!}}.{(\dfrac{1}{4})^3} + ....(1)$
Consider the expansion of${(1 - x)^{ - \dfrac{p}{q}}}$
\[{(1 - x)^{ - \dfrac{p}{q}}} = 1 + \dfrac{p}{{1!}}(\dfrac{x}{q}) + \dfrac{{p(p + q)}}{{2!}}{(\dfrac{x}{q})^2} + ....\]
Comparing (1) with the expansion, we get $p = 3$ and $p + q = 5$.
$ \Rightarrow q = 2$.
Also $\dfrac{x}{q} = \dfrac{1}{4} \Rightarrow x = \dfrac{1}{2}$
Substituting these values in the RHS of equation (1), we get
\[1 + S = {(1 - \dfrac{1}{2})^{ - \dfrac{3}{2}}} = {(\dfrac{1}{2})^{ - \dfrac{3}{2}}} = {(2)^{\dfrac{3}{2}}} = 2\sqrt 2 \]
Now, subtract 1 from both the sides to get S.
Therefore, $S = 2\sqrt 2 - 1$
That is, the sum of the given series is $S = 2\sqrt 2 - 1$.
Note: For any real number x such that$\left| x \right| < 1$ and rational number n, the binomial expansion of ${(1 + x)^n}$ is given by
${(1 + x)^n} = 1 + nx + \dfrac{{n(n - 1)}}{{2!}}{x^2} + .... + \dfrac{{n(n - 1)(n - 2)....(n - r + 1)}}{{r!}}{x^r} + .....$
Where $\dfrac{{n(n - 1)(n - 2)....(n - r + 1)}}{{r!}}$is the coefficient of the\[{r^{th}}\]term of the series
Complete step by step solution:
We have an infinite series $\dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....$
We need to find the sum of this series.
Let’s call the sum as S.
Then we have $S = \dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....(1)$
We will simplify the above expression by adding 1 on both the sides of the equation.
Thus, we have
$1 + S = 1 + \dfrac{3}{4} + \dfrac{{3.5}}{{4.8}} + \dfrac{{3.5.7}}{{4.8.12}} + ....$
Now, RHS can be expressed as follows:
$1 + S = 1 + \dfrac{3}{{1!}}.(\dfrac{1}{4}) + \dfrac{{3.5}}{{2!}}.{(\dfrac{1}{4})^2} + \dfrac{{3.5.7}}{{3!}}.{(\dfrac{1}{4})^3} + ....(1)$
Consider the expansion of${(1 - x)^{ - \dfrac{p}{q}}}$
\[{(1 - x)^{ - \dfrac{p}{q}}} = 1 + \dfrac{p}{{1!}}(\dfrac{x}{q}) + \dfrac{{p(p + q)}}{{2!}}{(\dfrac{x}{q})^2} + ....\]
Comparing (1) with the expansion, we get $p = 3$ and $p + q = 5$.
$ \Rightarrow q = 2$.
Also $\dfrac{x}{q} = \dfrac{1}{4} \Rightarrow x = \dfrac{1}{2}$
Substituting these values in the RHS of equation (1), we get
\[1 + S = {(1 - \dfrac{1}{2})^{ - \dfrac{3}{2}}} = {(\dfrac{1}{2})^{ - \dfrac{3}{2}}} = {(2)^{\dfrac{3}{2}}} = 2\sqrt 2 \]
Now, subtract 1 from both the sides to get S.
Therefore, $S = 2\sqrt 2 - 1$
That is, the sum of the given series is $S = 2\sqrt 2 - 1$.
Note: For any real number x such that$\left| x \right| < 1$ and rational number n, the binomial expansion of ${(1 + x)^n}$ is given by
${(1 + x)^n} = 1 + nx + \dfrac{{n(n - 1)}}{{2!}}{x^2} + .... + \dfrac{{n(n - 1)(n - 2)....(n - r + 1)}}{{r!}}{x^r} + .....$
Where $\dfrac{{n(n - 1)(n - 2)....(n - r + 1)}}{{r!}}$is the coefficient of the\[{r^{th}}\]term of the series
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE