Question #1d0d6

2 Answers
Tony B
Jul 6, 2017

#color(red)("Solution part 1 of 2")#

Also see part 2 of 2 for the calculation method

If you use a strait line of best fit you can not be precise enough to satisfactorily predict values.

Explanation:

Triangular numbers are constructed as in the diagram.
Tony B

You add up all the dots from the first one down to whichever point you wish to stop.

The #11^("th")# term works out to be 66

Tony B

Tony B
Jul 6, 2017

#color(red)("Solution part 2 of 2")# showing the quadratic

See part 1 of 2 first before reading this one.

Explanation:

The sequence is
#1" "3" "6" "10" "15" "21...#

#1 larr" 1st term"#

#1+2 = 3larr" 2nd term"#

#1+2+3=6 larr" 3rd term"#

#1+2+3+4=10larr" 4th term"#

#1+2+3+4+5=15larr" 5th term"#

#1+2+3+4+5+6=21larr" 6th term"#

Notice that the last value in the sum is the term number or line number.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider any one of the rows

Let the first value be #f# which in each case is 1

Let the last value be #L#

Then the sum of any row is #"count"xx"mean"#

count #=L# so we have:

#"count"xx"mean"" "->" "(f+L)/2xxL" "=" "f/2L+1/2L^2#

but #f=1# giving:

#1/2L+1/2L^2#

Changing the order we have:

#1/2L^2+1/2L+0#

Compare this to the standardised equation of a quadratic

#y=1/2L^2+1/2L+0#
#y=ax^2color(white)(.)+color(white)(.)bx+c#

so if #a_(11)# is the 11th term we have:

#a_11=1/2(11^2)+1/2(11) #

#a_11=60 1/2+ 5 1/2=66#

Tony B

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