Mathematics

Problem

A sequence of Riemann sums is defined by

S_n=\sum_{i=1}^{n}\bigl(k+2x_i\bigr)\,\Delta x,

where

x_i=\dfrac{3i}{n}

and

\Delta x=\dfrac{3}{n}

. It is known that

\displaystyle\lim_{n\to\infty}S_n=27

. (a) Write the definite integral represented by the limit of the sums

S_n

. Clearly state both the integrand and the interval of integration. (b) Determine the value of

k

and justify each step of your reasoning.