Answer to #24:-
You're right, JaneFairfax!

#29. Decompose into partial fractions:-

.

#30. If p and q are the roots of the equation x²+8x = -15, form the equation whose roots are (p+q) and 3pq.

Answer to #30:-
Excellent, JaneFairfax! You're right again!

When you worked out the area of the parallelogram, you use its slant height instead of its perpendicular height and so your answer is larger than the actual area.

I get the area of the trapezium to be the same as Jane's 3rd attempt.

#31. A person covers the distance from P to Q at the speed of 3 kilometers per hour. From Q to P, he covers the distance at 6 kilometers per hour. What is the average speed per hour?

Answer to #31:-
You are correct, JaneFairfax!

#34. Which is the largest number each number of the sequence is divisible by?

?
Why?

Answer to #34:
(1) The question has been modified. I didn't notice what meaning the question conveyed while posting, sorry!
(2) I have a proof that every number of the form

is divisible by 30.
I wanted to know if I am right.

This product is certainly divisible by 6.
For proving that

is always divisible by 5, I use this logic.

                           Last digit of

0                                                        0
1                                                        1
2                                                        6
3                                                        1
4                                                        6
5                                                        5
6                                                        6
7                                                        1
8                                                        6
9                                                        1

It can be seen that the last digit of

is always 0 or 5, which means it is a multiple of 5.
I hope my reasoning is acceptable!

Thanks, JaneFairfax!
That has made things a lot more clearer to me!

To JaneFairfax:
It occured to me that, as mentioned by you in post #70, if n is a multiple of of 5,

would not be divisible by 5.
That isn't going to affect the proof.
The idea is to prove that

is divisible by 30, hence, if

isn't divisible by 5, n would be. We already know that

is divisible by 2 and 3 (because (n-1)(n)(n+1) is a factor). Now that is has been shown that it is also divisible by 5, can we say decisively that a number of the form (n>0, n is a Natural number)

is divisible by 30 with the help of posts #69 and #75 alone, without the help of Fermat's little theorem?
Am I right?