math induction q (1 Viewer)

[vmoore]

ok, i usually have no probs with doing induction.. but this one q i have no idea what it means... i just dont get what its asking. The q is Q5a 1999 HSC and is:

Prove for all n positive integers:
(n + 1)(n + 2) ... (2n - 1)2n = 2^n[1x3x...x(2n-1)]

can someone explain what this notation atually means... i get the RHS but whats the deal with the LHS ?

 

[vds700]

ok, i usually have no probs with doing induction.. but this one q i have no idea what it means... i just dont get what its asking. The q is Q5a 1999 HSC and is:

Prove for all n positive integers:
(n + 1)(n + 2) ... (2n - 1)2n = 2^n[1x3x...x(2n-1)]

can someone explain what this notation atually means... i get the RHS but whats the deal with the LHS ?

The LHS is a product of terms in series Thats probably what threw you, they usually ask sum or series, divisibilty and inequalities

Assume true for n = k

(k+1)(k+2)...(2k-1)2k = 2^k[1x3x...x(2k-1)]

Prove true for n = k+1
(k+2)(k+3)...(2k+1)2(k+1) = 2^(k+1)[1x3x..x(2k+1)]

LHS = (k+1)(k+2)...(2k-1)(2k)2(2k+1) reararnging and putting the (k+3) into the ... and bringing out 2 terms preceding (2k+1) from the ...
=2^k[1x3x...x(2k-1)] 2(2k+1) putting in assumption
=2^(k+1)[1x3x...(2k-1)(2k+1)]
==2^(k+1)[1x3x...(2k+1)] putting the (2k-1) into the ....
=RHS

Hope this makes sense

 

[vmoore]

ah yes, it does make sense actually,
thanks... had never seen a q like that before.
ta