For every integer n ≥ 1, ( 3^2 ^ n -1) is always divisible by. - Vidyadip

MCQ

For every integer $\text{n} ≥ 1, ({3^2}^{\text{n}}-1)$ is always divisible by.

A
$2^{\text{n}^2}$
B
$2^{\text{n + 4}}$
✓
$2^{\text{n + 2}}$
D
$2^{\text{n + 3}}$

✓

Answer

Correct option: C.

$2^{\text{n + 2}}$

For $\text{n = 1},\ 3^{\text{2}^1}-1=8,$ which is divisible by $2^{\text{n + 2}}$
Let us assume that $3^{2^{\text{m}}}-1$ is divisible by $2^\text{m + 2}$ for some integral value of $m.$
Let us consider the expression for $m+1$
$3^{2^{\text{m+1}}}-1$
$=(3^{2^{\text{m}}}-1)\ \times (3^{2^{\text{m}}}+1)$
The first term is divisible $2^{\text{m+2}}$ and the second term is also an even number.
Hence, the term is divisible by $2^{\text{m+2}}$
Hence, by induction we can prove that it is true for all $m.$

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

Tangents are drawn from the point $(-1,-4)$ to the circle $x^2 + y^2 -2x + 4y + 1 = 0$. Length of corresponding chord of contact will be-

If $z = \frac{{\sqrt 3 }}{2} + \frac{i}{2}\,\,\,\left( {i = \sqrt { - 1} } \right)$, then ${\left( {1 + iz + {z^5} + i{z^8}} \right)^9}$ is equal to

Let $PQ$ be a focal chord of the parabola $y^{2}=4 x$ such that it subtends an angle of $\frac{\pi}{2}$ at the point $(3, 0)$. Let the line segment $PQ$ be also a focal chord of the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac{1}{e^{2}}$ is equal to

If $a$ and $c$ are positive real numbers and the ellipse $\frac{{{x^2}}}{{4{c^2}}} + \frac{{{y^2}}}{{{c^2}}} = 1$ has four distinct points in common with the circle $x^2 + y^2 = 9a^2$ , then

The value of $\mathop {\lim }\limits_{x \to - 1} \frac{{{x^2} + 3x + 2}}{{{x^2} + 4x + 3}}$ is equal to

If $A$ and $B$ are two given sets, then $\text{A}\cap\text{(A}\cap\text{B})^\text{c}$ is equal to:

The solution of the equation $2{x^2} + 3x - 9 \le 0$ is given by

Let $\mathrm{n}>2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is $99$ times the number of blue lines, then the value of $n$ is

Product of slopes of common tangents to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ and parabola $y^2 = 8x$ is -

The minimum value of $\cos \,2\theta \, + \,\cos \,\theta $ for real values $\theta $ is