Can anyone please help with these problems I have?
1. A particle moving in simple harmonic motion obeys the law:
d² x/dt² =-w² x
Show that d² x/dt² =v dv/dx, where v=dx/dt.

2. A sequence is defined by b₁ =1 and b_n+1 =b_n (b_n +1), for all n > = 1.
By using mathematical induction, prove that for each n, b_n is a positive integer and
$b_{n+1}=1+\sum _{k=1}^n{b}_k^2$.
3. Prove by mathematical induction that for n={N}, where p is a constant natural number, that 1 x 2 x 3 x ...p + 2 x 3 x 4 x ...p(p+1) +...+ n(n+1)(n+2)...(n+p-1)=[n(n+1)(n+2)...(n+p-1)(n+p)]/(p+1).

Thanks a lot!

regards..

Your first question has nothing to do with SHM. Well, it does apply in SHM, but in fact it is always true (not dependant on SHM).
d² x/dt² =d/dt(dx/dt) (by def'n of 2^nd derivative)
=dv/dt (since v=dx/dt)
=dv/dx . dx/dt (chain rule)
=v dv/dx (since v=dx/dt)

If it is true for $n=r-1$ then

$b_r=1+\sum _{k=1}^{r-1}{b}_k^2$

sub this into your formula for $b_{n+1}$, and work towards:

$b_{r+1}=1+\sum _{k=1}^r{b}_k^2$

then show that the formula works for $n=0$ (i.e. $b_1$) so that you have prooved the formula by induction. Use a similar approach for q.3.

Jim