practice problems pending

Q1.
Prove that:
1/15 <  1.3.5.7...99/2.4.6.8...100  < 1/10

S1.
We will use the property that
n/n+1 < n+1/n+2
i.e.
as the integer 'k' increases k/k+1 keeps becoming larger.

It is easy to prove it by cross multiplication:
n(n+2) < (n+1)^2 => 0 < 1.

Let's rewrite it as:
1/2 * 3/4 * 5/6 ... 99/100 = P
Let
Q = 2/3 * 4/5 ....98/99 * 100/101

Both P,Q have 50 fractions each.
And each fraction of P is less than its corresponding fraction in Q.

So P < Q
Multiply both sides by P.
P^2 < PQ = 1/2 * 2/3 * 3/4 .... 100/101 = 1/101
P^2 < 1/101 => P^2 < 1/100 => P < 1/10

Now for the other part where we have to prove that P > 1/15
P = 1/2 *(3/4 * 5/6 * ... 99/100)
R = 2/3 * 4/5 ... 98/99
=>
P > R/2
2P> R
2P^2 > P.R = 1/2 * 2/3 * 3/4 ... 99/100 = 1/100
P^2 > 1/200
P^2 > 1/225
P > 1/15
H.P.




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