The first three terms of an arithmetic sequence are u1=3k−1, u2=4k+1, and u3=6k−2. What is the value of k?

• From common difference of an arithmetic progression:

• substitute into the expression above:

let m = monomial

let t = trinomial

so, m times t = t.

monomial times a trinomial = another trinomial.

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.