Prove that 1 15 &lt; 1 2 ∗ 3 4 ∗ ⋯ ∗ 99 100 &lt; 1 10 My attempt:If we name the value A we have: A 2 &lt; ( 1 2 ∗ 3 4 ∗ ⋯ ∗ 99 100 ) ( 2 3 ∗ 4 5 ∗ … 100 101 ) = 1 101 ⇒ A &lt; 1 10 But I don't know, how to prove the other side?

Question

Prove that       1    15    &amp;lt;      1    2    ∗      3    4    ∗  ⋯  ∗      99    100    &amp;lt;      1    10  My attempt:If we name the value   A we have:      A    2    &amp;lt;      (          1      2        ∗          3      4        ∗    ⋯    ∗          99      100        )        (          2      3        ∗          4      5        ∗    …          100      101        )    =      1    101    ⇒  A  &amp;lt;      1    10  But I don&#039;t know, how to prove the other side?

Jaydon Roth · Accepted Answer

A    2    =      ∏          k      =      1              50                  (      2      k      −      1              )        2                    (      2      k              )        2              =      1    200        ∏          k      =      2              50                  (      2      k      −      1              )        2                    (      2      k      −      2      )      ⋅      (      2      k      )        &amp;gt;      1    200        ∏          k      =      2              50                  (      2      k      −      1              )        2                    (      2      k      −      1              )        2              =      1    200    &amp;gt;      1    225

Prove that 1/15<1/2∗3/4⋯99/100 <1/10

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