n (n+1) (n+5) is a multiple of 3 is true for - Vidyadip

MCQ

$n (n+1) (n+5)$ is a multiple of $3$ is true for

A
All natural numbers $n > 5$
B
Only natural number $3 ≤ n < 15$
✓
All natural numbers $n$
D
None

✓

Answer

Correct option: C.

All natural numbers $n$

Let the statement be denoted by $p(n)$
i.e., $P(n) : n (n+1) (n+5)$ is a multiple of $3$
For $n = 1, n(n+1) (n+5) = 1.2.6 = 12 = 3.4$
$P(n)$ is true for $n = 1$
Suppose $p(k)$ is true for $n = k$
i.e. $k(k+1) (k+5) =3m ($let$)$ or $k^3 + 6k^2 + 5k = 3m ... (i)$
Replacing $k$ by $k+1,$ we get
$(k+1) (k+2) (k+6) = k (k^2+ 8k + 12) + (k^2+ 8k + 12) $
$k^3+ 9k^2+ 20k + 12 = (k^3+ 6k^2+ 5k) + (3k^2+15k+12)$
$= 3m + 3k^2+ 15k + 12 [$from $(i)]$
$= 3 (m + k^2+ 5k + 4)$
i.e. $(k+1) (k+2) (k+6)$ is a multiple of $3$
i.e. $P(k+1)$ is multiple of $3,$ if $P(k)$ is a multiple of $3$
i.e. $P(k+1)$ is true whenever $P(k)$ is true.
Hence $P(n)$ is true for all $n \in N$

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

There are $n$ letters and $n$ addressed envelops. The probability that each letter takes place in right envelop is

$\frac{{{{( - 1 + i\sqrt 3 )}^{15}}}}{{{{(1 - i)}^{20}}}} + \frac{{{{( - 1 - i\sqrt 3 )}^{15}}}}{{{{(1 + i)}^{20}}}}$ is equal to

If slope of a line is negative then its inclination is:

If for positive integers $r> 1, n > 2$, the coefficients of the $(3r)^{th}$ and $(r + 2)^{th}$ powers of $x$ in the expansion of $( 1 + x)^{2n}$ are equal, then $n$ is equal to

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1=7$ and common difference $8$ . Let $T_1, T_2, T_3, \ldots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n \geq 1$. Then, which of the following is/are $TRUE$ ?

$(A)$ $T_{20}=1604$

$(B)$ $\sum_{ k =1}^{20} T_{ k }=10510$

$(C)$ $T_{30}=3454$

$(D)$ $\sum_{ k =1}^{30} T_{ k }=35610$

The point $(s) $ on the parabola $y^2 = 4x$ which are closest to the circle, $x^2 + y^2 - 24y + 128 = 0 $ is/are :

A bag contains $3$ red and $7$ black balls, two balls are taken out at random, without replacement. If the first ball taken out is red, then what is the probability that the second taken out ball is also red

The value $\cos 105^\circ + \sin 105^\circ $ is

Equation of the line which passes through the point $( - 4,\;3)$ and the portion of the line intercepted between the axes is divided internally in the ratio $5 : 3$ by this point, is

Given that $x, y$ and $b$ are real numbers and $x < y, b > 0,$ then: