Sum of arithmetic sequence inequality in answer

Question

I have been given the question :

5. Find the sum of the arithmetic progression: $$8.5 + 12 + 15.5 + 19 +\dotsb + 103$$

--for clarity there are $27$ terms, as $$\frac{103-8.5}{3.5} = 27$$
$n$ = (last term - 1st term)/difference

There are two methods to work this out :

1st method =

$$\frac{a_1 + a_n}{2} \cdot n$$

2nd method

$$(2a+(n-1)d) \cdot\frac{n}{2}$$

the 1st method is $$\frac{8.5 + 103}{2}\cdot 27 = 1505.25$$

the second method is $$((8.5\cdot2) + (26\cdot3.5))\cdot\frac{27}{2} = 1458$$

I can't understand why I get two different answers as both formulas are proven to be equal.

any help would be greatly appreciated.

Answer 1

To get from $8.5$ to $103$ requires $28$ terms, not $27$. The first approach is then $(8.5+103)*28/2=1561.$ The second is $(8.5\cdot 2+27\cdot 3.5)*28/2=1561$ You need $27$ increments of $3.5$, but that means you have $28$ terms. This is a fencepost error

Answer 2

The reason you had an error is that there is $28$ not $27$ numbers:

Why "$n+1$"? Is a question that bothered me for a bit but I finally got it.

A proof (with visuals):

If we plot your numbers we get something like this and some more quadrilaterals of course (numbers are off but I'm just showing you the shapes I'm talking about):

The main idea here is that $1$ of these special quadrilaterals outlined in blue has $2$ heights, and $2$ is one more than one. As we add these special shapes to each other side by side, one side is shared and one new one is introduced. Thus we have the recurrence relation:

$$a_1=2$$

$$a_{n+1}=a_n+1$$

And thus,

$$a_n=n+1$$

Meaning that if we have $n$ of these special quadrilaterals lined side by side, perfectly intersecting, we have $n+1$ heights which are perpendicular to the $x$ axis.

To add onto this, $n$ the number of rectangles, can be found by the formula which is often introduced in a Riemann sum:

$$n=\frac{b-a}{\Delta x}$$

Where $b$ is your last term/height and $a$ is your first. And $\Delta x$ is the difference between each consecutive heights.

Getting an intuition for this:

Most of us start counting with the number $1$. And we know that the sequence:

$1,2,3,4,...n$

Has $n$ terms as a result of the way we count. However the supposed thought of $n-1$ does not go with this, thus we add another $1$.