Express the sequence {a_n}_n = 1^infinity as an equivalent sequence of the form {b_n}_n = 3^infinity {n^2 + 7n - 5}_n = 1^infinity An equivalent sequence is { }_n = 3^infinity (Simplify your answer. DO not factor.)

We're given the sequence,

Since is quadratic, should also be quadratic. Replace with (if , then ) in the definition of , and let denote the new coefficients:

Expand the right side:

Coefficients of terms with the same degree should be the same:

So the new sequence is the same, with

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.