How do you use the properties of summation to evaluate the sum of \[\sum {i({i^2} + 1)} \]from i =1 to 10?

Hint: Here in this question, we have to find the summation of the given term. The summation is nothing but adding the terms of the sequence or the function. As in the question it is mentioned that the value of i ranges from 1 to 10. By substituting the value of i we determine the solution for the given question.

Complete step-by-step solution:
The summation is another name for addition. The numbers in a sequence should be added.
Now consider \[\sum {i({i^2} + 1)} \], here the value of i ranges from 1 to 10.
Therefore we have
\[ \Rightarrow \sum\limits_{i = 1}^{10} {i({i^2} + 1)} \]
Substituting the value of i from 1 to 10. While substituting the values for i the summation symbol will not be considered and written. So applying the summation we have
\[
   \Rightarrow 1({1^2} + 1) + 2({2^2} + 1) + 3({3^2} + 1) + 4({4^2} + 1) + 5({5^2} + 1) + 6({6^2} + 1) + 7({7^2} + 1) + \\
  8({8^2} + 1) + 9({9^2} + 1) + 10({10^2} + 1) \\
 \]
Squaring the terms which involves the term power 2.
\[
   \Rightarrow 1(1 + 1) + 2(4 + 1) + 3(9 + 1) + 4(16 + 1) + 5(25 + 1) + 6(36 + 1) + 7(49 + 1) + \\
  8(64 + 1) + 9(81 + 1) + 10(100 + 1) \\
 \]
Add the terms or the constants which are present in the braces. On adding the terms we have
\[ \Rightarrow 1(2) + 2(5) + 3(10) + 4(15) + 5(26) + 6(37) + 7(50) + 8(65) + 9(82) + 10(101)\]
If the term is in the braces and there is no arithmetic operation then we apply the multiplication arithmetic operation to the terms.
So on multiplying we get
\[ \Rightarrow 2 + 10 + 30 + 60 + 130 + 222 + 350 + 520 + 738 + 1010\]
Between the two terms there is a + symbol. This implies that there we use additional arithmetic operations. Therefore on adding we get
\[ \Rightarrow 3072\]
Hence by the properties of summation we have evaluated the summation.
Therefore \[ \Rightarrow \sum\limits_{i = 1}^{10} {i({i^2} + 1)} = 3072\]


Note: The summation symbol is usually represented by \[\sum\limits_{}^{} {} \]. The summation is as defined as \[\sum\limits_{k = 1}^n {{a_k}} = {a_1} + {a_2} + {a_3} + ... + {a_n}\], this the main property of summation. We obtain the sequence and then by adding the sequence we obtain the value. The sequence is written on some rule.