Mathematics Test 221

Let n, n + 1, n + 2 be three consecutive numbers.

Then, P(n) = n³ + (n + 1)³ + (n + 2)³

We prove by induction on n.

P(1) = 1 + 8 + 27 = 36, which is divisible by 9.

Let the result be true for k, i.e. P(k) = k³ + (k + 1)³ + (k + 2)³ is divisible by 9.

For k + 1, we have

P(k + 1) = (k + 1)³ + (k + 2)³ + (k + 3)³

= (k + 1)³ + (k + 2)³ + k³ + 9k² + 27k + 27

= k³ + (k +1)³ + (k + 2)³ + 9(k² + 3k + 3) = P(k) + 9(k² + 3k + 3), which is divisible by 9.

Hence, P(n) is divisible by 9 for all natural numbers.

nth term = n(n + 1)(n + 2) = n(n² + 3n + 2) = n³ + 3n² + 2n. Let the sum be S, then

S = [{n(n + 1)/(2)}² + 3n(n + 1)(2n + 1)/6 + 2n(n + 1)/2]

S = n(n + 1)[n(n + 1)/4 + (2n + 1)/2 + (1)]

S = n(n + 1)[(n² + n + 4n + 2 + 4)/4]

S = n(n + 1)(n + 2)(n + 3)/4

Let PQ be the focal chord of the parabola. Let the coordinates of P and Q be (at₁², 2at₁) and (at₂², 2at₂), respectively.

Then, t₁t₂ = -1

Let (h, k) be coordinates of the mid-point of PQ. Then,

As the line lies in the plane, the point (2, 1, -2) lies in the plane and normal to the plane is perpendicular to the line.

So, 2 + 3(1) - α(-2) + β = 0 and (3)(1) + (3)(-5) - α(2) = 0

α = -6 and β = 7

The given statement is, 'If I become a teacher, then I will open a school'.

Negation of the given statement is, 'I will become a teacher and I will not open a school'.

( -(p → q) = p ^ - q)

This is equal to zero only if n + 2 is odd i.e. n is odd integer.