2 questions: angle b/w two lines and enequality induction. (1 Viewer)

sinophile · Feb 2, 2009

1. find the acute angle b/w the pair of two straight lines with gradients 4 and 3/5.

2. prove (2^n)>(n^2) for n>4

lychnobity · Feb 2, 2009

sinophile said:
1. find the acute angle b/w the pair of two straight lines with gradients 4 and 3/5.

2. prove (2^n)>(n^2) for n>4

1) let @ be the acute angle

tan @ = | (m1 - m2) / (1 - m1m2) |
= |(4 - 0.6)/ (1 - 3.4)|
=17/12

@ = tan^-1 (17/12)
@ = 55 degrees

Trebla · Feb 3, 2009

sinophile said:
2. prove (2^n)>(n^2) for n>4

$$When $ n = 5 $\\ LHS=2^5\\ = 32\\ RHS = 5^2\\ = 25\\ LHS > RHS\, \therefore\,$Statement is true for n = 5$\\ $Assume the statement is true for $n = k\\ 2^k > k^2\\ $Required to prove the statement is true for $n = k + 1\\ 2^{k+1} > (k+1)^2\\ LHS = 2^{k+1}\\ =2.\,2^k\\ >2k^2 $ by assumption$\\ = k^2 + k^2\\ $But $k^2 > 2k + 1$ for $ k > 4\\ $(Sketch a graph of $y = x^2$ and $y = 2x + 1$ to see this or alternatively note that $k^2>16$ and $2k>8$ for $k>4$ hence $k^2-2k>8>1)\\$So continuing the proof$\\ > k^2+2k+1\\ =(k+1)^2\\ = RHS\\ $The statement is true for $n=k+1$, if it is true for $n=k$. Since it is true for $n=5$, it follows by induction that it is true for all integers $n>4$.$

2 questions: angle b/w two lines and enequality induction. (1 Viewer)

sinophile

Well-Known Member

lychnobity

Active Member

Trebla

Administrator

Users Who Are Viewing This Thread (Users: 0, Guests: 1)