Question

Question #1d0d6

Answer 1

`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.

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

Answer 2

`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`