Question 933569
# of chairs in rows: 20, 22,24,26,28,30,32,34,36,38,40,42,44,46,48,50

difference {{{d=2}}}
{{{a[1]=20}}}

{{{a[n] = a[1] + (n-1)d}}}



a.
 how many chairs will be in the 25th row?

{{{a[25 ]= 20 + (25-1)2}}}

{{{ a[25 ]=20+50-2}}}

{{{a[25 ]=68}}}...chairs will be in the 25th row

b.
with {{{1000}}} chairs  how many perfect rows can you make


 {{{1000= 20 + (n-1)2}}}
 {{{1000-20 = (n-1)2}}}
{{{980 = (n-1)2}}}
{{{980 /2= (n-1)}}}
{{{490= n -1}}}
{{{491= n}}} .....# of  rows  you can make, since {{{491}}} is NOT a perfect number, 

the only perfect numbers between {{{1}}} and {{{1000}}} are {{{6}}}, {{{28}}}, {{{496}}}; since {{{496>491}}} ,  with {{{1000}}} chairs  you can  make only {{{two}}} perfect rows which are rows {{{6}}} and{{{28}}}

c.

how many chairs will be in the last row? 
 the last row is {{{a[n]}}}:

{{{a[n ]= 20 + (n-1)2}}} ...# chairs in the last row

d.


what are the difference in the number of chairs for arranging 25 rows and 50 rows? 
{{{a[25 ]=68}}}...chairs will be in the 25th row

{{{a[50 ]= 20 + (50-1)2}}}

{{{a[50 ]= 20 + (49)2}}}

{{{a[50 ]= 20 + 98}}}

{{{a[50 ]= 118}}}

 the difference in the number of chairs for arranging {{{25}}} rows and {{{50}}} rows is {{{118-68=50}}}

or, since  the difference in the number of chairs for arranging between two rows  is 2 chairs, then
between {{{25}}} rows and {{{50}}} will be {{{25*2=50}}}

e.
in which row {{{50}}} chairs will exist? 

{{{50 = 20 + (n-1)2}}}

{{{50 - 20 = (n-1)2}}}

{{{30/2 = (n-1)}}}

{{{15+1= n}}}

{{{n=16}}} .....row in which  {{{50}}} chairs will exist

f.
will you be able to make perfect rows with thousand chairs? if not how many chairs are needed to make the row perfect? 

from c. we know that 

{{{a[491 ]=1000 }}} and {{{491}}} is not  perfect row because {{{491}}} is NOT a perfect number
first perfect number is {{{n=496}}}(see b.); so we need 5more rows or 10 more chairs
 
{{{a[496]= 20 + (496-1)2}}}
{{{a[496]=20 +(495)2}}}
{{{a[496 ]-100=20+990}}}
{{{a[496 ]=1010}}} 


g.

if you need to add {{{2}}} more rows you will have {{{498}}} rows

as given,for each additional row you need {{{2}}} more chair; so,  to make {{{2}}} more rows  you will need {{{4}}} more chairs to row {{{496}}} which is {{{1010}}}; so, we will have {{{1014}}}  chairs in row # {{{498}}}

check:
{{{1014= 20 + (n-1)2}}}

{{{1014= 20 + (n-1)2}}}

{{{1014-20 =2n-2 }}}

{{{994+2=2n}}}

{{{996=2n}}}

{{{996/2=n}}}

{{{n=498}}}...so, we got a row number {{{498}}}