Question 1168309
.
The sum of 128 consecutive odd, positive integers is the greatest power of two less than 200,000. 
The sum of the least and the greatest of these integers is 2^N. What is the value of N?
~~~~~~~~~~~~~~~



<pre>
{{{log(2, (200000))}}} = 17.61 (approximately, with two decimals).


Therefore, the greatest power of 2 less than 200,000 is  {{{2^17}}} = 131072.


So, the sum of 128 consecutive odd positive integers is  131072.


These consecutive integer numbers form an arithmetic progression;

therefore, their sum is the product of the number of terms (128) by half the sum of the least and the greatest of these two numbers.


In other words,  {{{2^17}}} = {{{128*(1/2)*2^N}}}.


It is equivalent to


    {{{2^17}}} = {{{2^7*2^(-1)*2^N}}},  or

    {{{2^17}}} = {{{2^(7-1+N)}}},       or

    {{{2^17}}} = {{{2^(N+6)}}}.


It implies  17 = N + 6,  or  N = 17 - 6 = 11.


<U>ANSWER</U>.  N = 11.
</pre>

Solved.