Use the properties of summation and Theorem 4.2 to evaluate the sum.
A)4411
B)77,275
C)4319
D)65,016
E)77,259

Question

Use the properties of summation and Theorem 4.2 to evaluate the sum. 
A)4411 
B)77,275 
C)4319 
D)65,016 
E)77,259

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