Maths induction.... (1 Viewer)

[WouldbeDoctor]

I realise it is late but I am greatly need of some maths assistance, i don't expect this to be answer right now... here goes.

1.

By using mathematical induction prove the following relation, where n is greater or equal to 1

6( 1^2 + 2^2 + 3^3...... n^2) = n(n+1)(2n+1)

2. A motor bike starting from rest has an acceleration a m/s^2 given as a function of t seconds by:

a= 3/(2 square root --> (t+1)) and (0 is less than or equal to t which is less than or equal to 120)

Find i.) the velocity fuction and

ii) the displacement function of the motorbike

iii) what are the motor bike's position and speed one minute after it started.

 

[Sober]

By using mathematical induction prove the following relation, where n is greater or equal to 1

6( 1^2 + 2^2 + 3^3...... n^2) = n(n+1)(2n+1)

Prove true for n=1:

LHS = 6(1²) = 6

RHS = 1(2)(3) = 6 = LHS

Assume true for n=k:

6( 1² + 2² + 3² ... k²) = k(k+1)(2k+1)

Prove true for n=k+1:

6( 1² + 2² + 3² ... k² + (k+1)²) = (k+1)(k+2)(2k+3)

LHS = 6( 1² + 2² + 3² ... k²) + 6 (k+1)²

= k(k+1)(2k+1) + 6 (k+1)² (from the induction hypothesis)

= 2k³ + 9k² + 13k + 6

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

= RHS

~~~~~~~~~

It's late, I might do the other tomorrow.