1+frac{2}{4}+frac{2 cdot 5}{4 cdot 8}+frac{2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $S=1+\frac{2}{4}+\frac{2 \cdot 5}{4 \cdot 8}+\frac{2 \cdot 5 \cdot 8}{4 \cdot 8 \cdot 12}+\ldots$Comparing this with the binomial expansion $(

Vedclass · Accepted Answer

$\sqrt[3]{16}$

Vedclass · Answer

(B) Let $S=1+\frac{2}{4}+\frac{2 \cdot 5}{4 \cdot 8}+\frac{2 \cdot 5 \cdot 8}{4 \cdot 8 \cdot 12}+\ldots$Comparing this with the binomial expansion $(1-x)^n = 1 + n(-x) + \frac{n(n-1)}{2!}(-x)^2 + \ldots$We have $nx = \frac{2}{4} = \frac{1}{2}$ $(i)$And $\frac{n(n-1)}{2!}x^2 = \frac{2 \cdot 5}{4 \cdot 8} = \frac{10}{32} = \frac{5}{16}$ (ii)Dividing (ii) by the square of $(i)$: $\frac{\frac{n(n-1)}{2}x^2}{n^2x^2} = \frac{5/16}{1/4} \Rightarrow \frac{n-1}{2n} = \frac{5}{4}$$4n - 4 = 10n$ $\Rightarrow -6n = 4$ $\Rightarrow n = -2/3$Substituting $n$ into $(i)$: $(-2/3)x = 1/2 \Rightarrow x = -3/4$Thus,$S = (1 - x)^n = (1 - (-3/4))^{-2/3} = (7/4)^{-2/3}$ is incorrect. Re-evaluating: The series is $(1-x)^n$ where $nx = 1/2$ and $x = 1/2$ is not correct. Using $(1-x)^{-n} = 1 + nx + \frac{n(n+1)}{2!}x^2 + \ldots$Here $nx = 1/2$ and $\frac{n(n+1)}{2}x^2 = 5/16$. Solving gives $n=2/3$ and $x=3/4$.$S = (1-3/4)^{-2/3} = (1/4)^{-2/3} = 4^{2/3} = \sqrt[3]{16}$.

$1+\frac{2}{4}+\frac{2 \cdot 5}{4 \cdot 8}+\frac{2 \cdot 5 \cdot 8}{4 \cdot 8 \cdot 12}+\frac{2 \cdot 5 \cdot 8 \cdot 11}{4 \cdot 8 \cdot 12 \cdot 16}+\ldots \ldots$ is equal to :

Similar Questions

The property utilized in the manufacture of lead shots is ...............

Which of the following acts as both an oxidizing agent and a reducing agent?

$A$ conducting rod $AC$ of length $4l$ is rotated about a point $O$ in a uniform magnetic field $\vec B$ directed into the paper. $AO = l$ and $OC = 3l$. Then:

Two sources of equal emf $E$ are connected in series to an external resistance $R$. The internal resistances of the two sources are $R_1$ and $R_2$ $(R_2 > R_1)$. If the potential difference across the source having internal resistance $R_2$ is zero,then the value of $R$ is:

An object takes $n$ times as much time to slide down a $45^{\circ}$ rough inclined plane as it takes to slide down a perfectly smooth inclined plane of the same inclination. The coefficient of kinetic friction between the object and the rough incline is given by:

Vedclass Test Series

Exam Paper Generator

Online Exam Module