Question 1158294

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Problem 1....<br>
I will ignore the last number in the sequence you show, since it is almost certainly a typo.<br>
(By the way, it would be polite of you to make sure the problem is correct before you post it....)<br>
So I will use the first five of the terms you show and use those to find the correct 6th term and the new 7th term.<br><pre>
     17    15    25    53    105
        -2    10    28    52
           12    18    24
               6     6</pre>

The row of third differences is constant.  To find two more terms in the sequence, add two more 6's in that row and work back up the array.<br><pre>
     17    15    25    53    105    187   305
        -2    10    28    52     82    118
           12    18    24     30    36
               6     6     6     6</pre>


we find the 3rd differences to be all equal to {{{ 6}}}
so, the next 2nd difference will be {{{30+6=36}}}
the next 1st difference will be {{{82+36=118}}}
thus, the next term will be {{{187+118=305}}}



Problem 2....<br>
{{{a[1]=2}}}; {{{a[2]=4}}}; {{{a[3]=6}}}; {{{a[4]=8}}}; {{{a[5]=10}}}...<br>
Clearly the nth term is {{{a[n]=2n}}}; then {{{a[30]=2*30 = 60}}}<br>


Problem 3....<br>
(Note your statement of the problem is not complete....)<br>
The numbers of regions for 1, 2, 3, and 4 cuts are<br>
2, 4, 7, 11<br>
Compare this sequence to the sequence of triangular numbers:<br>
1, 3, 6, 10<br>
The numbers in this sequence are each 1 more than the corresponding triangular number.<br>
The formula for the n-th triangular number is {{{(n(n+1))/2}}}<br>
The formula for the maximum number of pieces of pizza with n cuts is {{{(n(n+1))/2+1}}}<br>