If p(n): 49^n+16^ n is divisible by 64 for n is true, then the le… - Vidyadip

MCQ

If $p(n): 49^\text{n}+16^{\text{n}}\lambda$ is divisible by $64$ for $\text{n}\in\text{N}$ is true, then the least negative integral value of $\lambda$ is:

A
$-3$
B
$-2$
✓
$-1$
D
$-4$

✓

Answer

Correct option: C.

$-1$

$ (49)^n+16 n-1 $
$ \Rightarrow(1+48)^n+16 n-1 $
$ \Rightarrow 1+48 n+\ldots 48^n+16 n-1 $
$ \Rightarrow 64 n+n C_2(48)^2+n C_3(48)^3+\ldots+(48)^n $
$ \Rightarrow 64\left(n+n C_2(6)^2+n C_3(6)^3 48+\ldots+(6)^n 8^{n-2}\right)$
$\therefore 49^n+ 16n - 1$ is divisible by $64$

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 "M.C.Q (1 Marks)" from Principle of Mathematical Induction→Principle of Mathematical Induction — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

In how many ways a team of $10$ players out of $22$ players can be made if $6$ particular players are always to be included and $4$ particular players are always excluded

If the coefficients of ${x^2}$ and ${x^3}$ in the expansion of ${(3 + ax)^9}$ are the same, then the value of $a$ is

A man walks a distance of $3$ units from the origin towards the north-east $\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)$ direction. From there, he walks a distance of $4$ units towards the north-west $\left(\mathrm{N} 45^{\circ} \mathrm{W}\right)$ direction to reach a point $\mathrm{P}$. Then the position of $\mathrm{P}$ in the Argand plane is

The sum of squares of deviations for $10$ observations taken from mean $50$ is $250$. The co-efficient of variation is.....$\%$

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is

For the circle ${x^2} + {y^2} + 6x - 8y + 9 = 0$, which of the following statements is true

Let a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}, k>0$, touch the $x$-axis at $(1,0)$. If the line $x + y =0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$ , then the value of $h + k + r$ is equal to

If $A, B$ and $C$ are any three sets, then $\text{A}-(\text{B }\cup\text{ C})$ is equal to.

Area of the circle in which a chord of length $\sqrt 2 $ makes an angle $\frac{\pi }{2}$ at the centre is

A relation $\phi$ from $C$ to $R$ is defined by $\text{x }\phi\text{ y}\Leftrightarrow|\text{x}|=\text{y}.$ Which one is correct?