Value of ^ _ k=1 ^ k _ r=0 1 3^ k ( ^k C_ r ) is: - Vidyadip

MCQ

Value of $\sum^{\infty}_{\text{k}=1}\sum^{\text{k}}_{\text{r}=0}\frac{1}{3^{\text{k}}}\big({^\text{k}}\text{C}_{\text{r}}\big)$ is:

✓
$2$
B
$\frac{2}{3}$
C
$\frac{1}{3}$
D
$\text{None of these}$

✓

Answer

Correct option: A.

$2$

$\sum\frac{1}{3^{\text{k}}}{^\text{k}}\text{C}_{\text{r}}$
$=\frac{1}{3^{\text{k}}}\sum{^\text{k}}\text{C}_{\text{r}}$
$=\frac{2^{\text{k}}}{3^{\text{k}}}$
This is a G.P
Therefore, the sum of the series will be
$\text{S}=\frac{\frac{2}{3}}{1-\frac{20}{3}}=2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from BINOMIAL THEOREM→BINOMIAL THEOREM — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If the circle ${x^2} + {y^2} = {a^2}$ cuts off a chord of length $2b$ from the line $y = mx + c$, then

Consider a cuboid all of whose edges are integers and whose base is a square. Suppose the sum of all its edges is numerically equal to the sum of the areas of all its six faces. Then, the sum of all its edges is

The area (in sq. units) of the region $\mathrm{S}=\{\mathrm{z} \in \mathbb{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{lm}(\mathrm{z}) \geq 0\}$ is

If $(a + ib)(c + id)(e + if)(g + ih)$$ = A + iB,$ then $({a^2} + {b^2})({c^2} + {d^2})({e^2} + {f^2})({g^2} + {h^2})$ =

If the roots of the given equation $({m^2} + 1){x^2} + 2amx + {a^2} - {b^2} = 0$ be equal, then

If the fourth term in the expansion of $\left(x+x^{\log _{2} x}\right)^{7}$ is $4480,$ then the value of $x$ where $x \in N$ is equal to

If the parabola ${y^2} = 4ax$ passes through the point $(1, -2)$, then the tangent at this point is

Let ${z_1},{z_2}$ be two complex numbers such that ${z_1} + {z_2}$ and ${z_1}{z_2}$ both are real, then

If $\text{|x}+2|\leq9,$ then:

If $a,\,b,\,c$ are in $A.P.$, then $(a + 2b - c)$ $(2b + c - a)$ $(c + a - b)$ equals