What conjecture can you make about the sum of the first 10 odd numbers?

The sum 10 first odd numbers are:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

You can see that the unit part of the numbers repeat, but in the latter 5 numbers we have:

1 + 3 + 5 + 7 + 9 + (10 + 1) + (10 + 3) + (10 + 5) + (10 + 7) + (10 + 9)

Now, we can rearrange this as:

2*(1 + 3 + 5 + 7 + 9) + 5*10

2*(23) + 5*10

now, every number multiplied by a 2, is an even number, and 5*10 = 50 is also an even number.

Here we can see that the sum of the first 10 odd numbers is an even number, and because is an even number different than 2, it is not a prime number.

You can solve the multiplications and get:

2*23 + 5*10 = 46 + 50 = 96

where are the choices?

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.