Use the properties of summation and Theorem 4.2 to evaluate the sum.

Question

Quizplus · Accepted Answer

Using Theorem 4.2, we can rewrite the sum as:

$\sum\limits_{i=1}^{10} (i+2)^2 =  \sum\limits_{i=1}^{10} (i^2+4i+4)$

Using the properties of summation, we can split this into three separate sums and evaluate each one:

$\sum\limits_{i=1}^{10} i^2 + \sum\limits_{i=1}^{10} 4i + \sum\limits_{i=1}^{10} 4$

Using the formula for the sum of the first $n$ squares, we have:

$\frac{n(n+1)(2n+1)}{6}$

So,

$\sum\limits_{i=1}^{10} i^2 = \frac{10(10+1)(2(10)+1)}{6} = 385$

Using the formula for the sum of an arithmetic series, we have:

$\frac{n}{2}(a_1 + a_n)$

where $a_1$ is the first term, $a_n$ is the $n$th term, and $n$ is the number of terms. So,

$\sum\limits_{i=1}^{10} 4i = \frac{10}{2}(4+4(10)) = 220$

And,

$\sum\limits_{i=1}^{10} 4 = 4(10) = 40$

Adding these three separate sums, we get:

$\sum\limits_{i=1}^{10} (i+2)^2 = 385 + 220 + 40 = 645$

Therefore, the correct answer is B)77275.

Use the Properties of Summation and Theorem 4