Question

Evaluate \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{2}\)

Short answer

The limit of the given expression is 182.

Step-by-step solution

Identify the Riemann sum

The given expression, \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{2}\), resembles the Riemann sum of an integral of a function. Let's rewrite the given expression in terms of a Riemann sum: $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i) \Delta x $$ where \(f(x_i) = \left(5 + \frac{2i}{n}\right)^2\), \(\Delta x = \frac{2}{n}\), and \(x_i\) represents the i-th term in the sum. In this case, \(x_i = a + i \Delta x\), where \(a\) is the left endpoint of the interval.

Find the function, interval, and left endpoint

Let's identify the function, interval, and left endpoint for our problem. From the expression of \(f(x_i)\), we get the function: $$ f(x) = (5 + x)^2 $$ Since \(\Delta x = \frac{2}{n}\), and we have a sum from \(i = 1\) to \(i = n\), we can conclude that the interval of integration is \([a, a+2]\). We can find the left endpoint (\(a\)) by finding the term when \(i = 1\): $$ x_1 = a + 1 \cdot \frac{2}{n} = 5 + \frac{2(1)}{n} \Rightarrow a = 5 $$ Now we have the function \(f(x) = (5 + x)^2\), the interval \([5, 7]\), and the left endpoint \(a = 5\).

Write the Riemann sum as a definite integral

Now, using the function and interval, we can rewrite the Riemann sum as a definite integral: $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i) \Delta x = \int_{5}^{7} (5 + x)^2 dx $$

Evaluate the definite integral

We need to evaluate the integral \(\int_{5}^{7} (5 + x)^2 dx\). Let's first expand the integrand: $$ (5 + x)^2 = 25 + 10x + x^2 $$ Now, find the antiderivative with respect to x: $$ \int (25 + 10x + x^2) dx = 25x + 5x^2 + \frac{1}{3}x^3 + C $$ Finally, evaluate the antiderivative at the endpoints of the interval and find the definite integral: $$ \int_{5}^{7} (5 + x)^2 dx = \left[25x + 5x^2 + \frac{1}{3}x^3\right]_5^7 = \left[25(7) + 5(7)^2 + \frac{1}{3}(7)^3\right] - \left[25(5) + 5(5)^2 + \frac{1}{3}(5)^3\right] = 182 $$

Write the final answer

Now that we've evaluated the definite integral, we have our limit: $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{2} = 182 $$