I think that if i put the picture on the web it would be clearer than what i type
for Q1: http://tinypic.com/r/2mqsq5c/5
(I need help in the making a conjecture for Vn using the fact that: 3=1+2 ...)
for Q2: http://tinypic.com/r/5z47qc/5
(I need help in making a conjecture of Un )
Thanks
Hi,
Please show any working you have. It can't be that you cannot even write the terms of both sequences. Thanks.
Cheers,
Wen Shih
to find a formula for Vn, you need to work out the first few terms of Vn, not just rely on the Un.
if you do so, you will find Vn = (n+2)/2^n
for Q2, if you had found U2 and U3, you would have found that for each term, the numerator is the sum of a GP series of n+1 terms, and the denominator is the sum of the same GP series but only of n terms. use GP Sn formula to get
Un = S(n+1)/S(n) = (1 - e^(n+1))/(1-e^n)
Dear Tvtokyo and students,
In O-level Elementary Mathematics, you would have been exposed to conjectures in the form of number patterns. Try to recall some techniques your school has taught you.
In addition, one ought to be exposed to common patterns (not explicitly taught or elaborated in detail in schools unfortunately) which I will elaborate with specific examples:
Pattern 1: 1, 3, 5, 7, ..., 2n + 1 or 2n - 1, ...
This is an odd number sequence.
Pattern 2: 0, 2, 4, 6, ..., 2n, ...
This is an even number sequence.
Pattern 3: 1, 4, 7, 10, ..., 3n - 2, ...
This is an arithmetic sequence.
Pattern 4: -1, 1/2, -1/4, 1/8, ..., (-1)(-1/2)^{n - 1}, ...
This is an alternating geometric sequence.
Pattern 5: 1, 4, 9, 16, 25, ..., n^2, ...
This is a sequence of perfect squares.
Pattern 6: 1, 2, 6, 24, 120, 720, ..., n!, ...
This is a sequence of factorials.
Pattern 7: 1/3, 3/5, 5/7, 7/9, 9/11, ..., (2n + 1)/(2n + 3) or (2n -1)/(2n + 1), ...
In this sequence, the numerators are odd numbers and denominators are 2 more than numerators.
Pattern 8: 2, 2/3, 2/9, 2/27, 2/81, ..., 2 / 3^{n - 1}, ...
In this sequence, the numerator is always a constant 2 and denominators are powers of 3, with each power being 1 less than the sequence number.
Pattern 9: 1, 3, 6, 10, 15, 21, 28, ..., 1/2 n(n + 1), ...
In this sequence:
1 is the sum of 1 term;
3 is the sum of 2 terms, i.e., 1 and 2;
6 is the sum of 3 terms, i.e., 1, 2 and 3;
10 is the sum of 4 terms, i.e., 1, 2, 3 and 4;
and so on...
Thanks.
Cheers,
Wen Shih