State which of the following statements is true. For every integer $n, n (n + 1) (n + 2)$ is divisible by

7

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6

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12

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Solution

The correct option is C $6$
$T h e g i v e n e x p r e s s i o n i s f (x) = n (n + 1) (n + 2) N o w w h e n n = 1 \to f (n) = 1 \times 2 \times 3 d i v i s i b l e b y 6 n = 2 \to f (n) = 2 \times 3 \times 4 d i v i s i b l e b y 6 = 6 \times 4 n = 3 \to f (n) = 3 \times 4 \times 5 d i v i s i b l e b y 6 ∴ B y i n d u c t i o n w e c a n s a y n \times (n + 1) \times (n + 2) w i l l b e d i v i s i b l e b y 6 f o r a l l i n t e g r a l v a l u e s o f 6. T h e s t a t e m e n t i s t r u e b e c a u s e 3 c o n s e q u t i v e i n t e g e r s a l w a y s c o n t a i n 2 a n d 3 a s f a c t o r s o f a n y t w o n u m b e r s o u t o f 3. S o t h i s t y p e o f t h e p r o d u c t o f c o n s e q u t i v e n u m b e r s i s a l w a y s d i v i s i b l e b y 6.$

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Similar questions

n \geq 2,

\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + . . . + \frac{1}{\sqrt{n}} > \sqrt{n}

\sqrt{n (n + 1)} < n + 1

P (n)

n (n + 1) (n + 2)

6

P (3)

Prove that n2 -n is divisible by 2 for every positive integer n

1^{n} + 8^{n} - 3^{n} - 6^{n}

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