Applicant 2 2/3 pages in five minutes how many pages per hour

  "32 pages [per hour] " .
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Explanation:
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Set up a ratio:
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2 ⅔ pages : 5 minute:: x pages: 1 hour ;  Solve for "x" ;

Rewrite as:
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(2 ⅔ pages) / 5 minutes = "x" pages/ 1 hour.

Cross-multiply ; and write as:

(5 minutes) ("x" pages) = (2 ⅔ pages) / 1 hour. ; 
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Convert the "5 min"  to its values in units of "hour" ;
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   5 min = ? hour ?
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   5 min * (  = ( hr ;

( hr  = [(5÷5)/(60÷5)] hr = (1/12) hr.
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So we take our ratio, which is in the form of an equation, as follows:
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                 (5 minutes) ("x" pages) = (2 ⅔ pages) * 1 hour.
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       and rewrite — substituting "(1/12 hr)" in lieu of "(5 min)" — as follows:
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      (1/12) * ("x" pages) = (2 ⅔ pages) * (1) ;  Solve for "x" ;
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          Divide EACH SIDE of the equation by:  "(1/12 hr)"; or, more efficiently, multiply EACH SIDE of the equation by "12 hr" ; since "dividing by "(1/12)" is the same as multiplying by "12" ;

(i.e. on the "left hand side" of the equation:

[ (1/12) x ] / (1/12) = x ;

  AND:  (1/12)x * 12  = (1x/12) * (12/1) = (12x/12) = x ; the same result.
            {or: (1/12)x * 12 = (x/12) * (12/1) = ?  ;  The "12's" cancel out to "1's" ;  
                        since: "12÷12 = 1" ;  and we have:
                                    " (x/1) * (1/1) = x * 1 = x ;  which is the same value.
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   to isolate "x" on one side of the equation:
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          So, as explained above, we multiply EACH SIDE of the equation by "12" ; as follows:
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  [ (1/12 hr) * ("x") ]  * 12  =  (2 ⅔ pages) * (1) * (12)  ;
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Simplify:
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                  x = (2 ⅔ pages) * (12) ; 

                  x = (8/3) * (12/1) = ?

The "3" cancels to a "1" ; and the "12" cancels to a "4" ;
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since: "12÷3 = 4" ;  and since:  "3÷3 = 1 ".
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  So rewrite as:
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              x = (8/1) * (4/1) = 8 * 4 = 32 pages.
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Alternate:
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     When we have:  x = (2 ⅔ pages) * (12) ; 

                 x = (8/3) * (12/1) = (8*12)/ (3*1) = 96/3 = 32 .
   
                 x = 32 pages.
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Alternate:
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     When we have:  x = (2 ⅔ pages) * (12) ; 

                 x = (8/3) * (12/1) = (8*12)/ (3*1) ;

                 x = (8*12)/ (3*1) ;
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We can cancel out the "12 in the numerator" and the "3" in the denominator, and replace with "4" and "1", respectively; and rewrite as:

                 x = (8*4) / (1*1) = (32/1) = 32.
  
                 x = 32  pages per hour.
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, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for