Let a subscript n equals n squared minus 3 n plus 3 comma space n greater or equal than 1. Find stack sum space a subscript i with i equals 1 below and 4 on top.

The given sequence is \( a_n = n^2 - 3n + 3 \) where \( n \geq 1 \). You are asked to find the sum of the sequence for \( i \) running from 1 to 4, i.e.,

\[
\sum_{i=1}^{4} a_i
\]

### Step-by-step Calculation

We will evaluate \( a_i \) for \( i = 1, 2, 3, 4 \).

1. For \( i = 1 \):
   \[
   a_1 = 1^2 - 3(1) + 3 = 1 - 3 + 3 = 1
   \]

2. For \( i = 2 \):
   \[
   a_2 = 2^2 - 3(2) + 3 = 4 - 6 + 3 = 1
   \]

3. For \( i = 3 \):
   \[
   a_3 = 3^2 - 3(3) + 3 = 9 - 9 + 3 = 3
   \]

4. For \( i = 4 \):
   \[
   a_4 = 4^2 - 3(4) + 3 = 16 - 12 + 3 = 7
   \]

Now, summing these values:

\[
\sum_{i=1}^{4} a_i = a_1 + a_2 + a_3 + a_4 = 1 + 1 + 3 + 7 = 12
\]

### Final Answer:
\[
\sum_{i=1}^{4} a_i = 12
\]

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Would you like a more detailed explanation or any additional clarifications?

### Related questions:
1. How do you find the sum of a sequence for a general expression \( a_n = n^2 - 3n + 3 \) for a given range?
2. What is the general method for calculating a sum of a sequence from \( i = 1 \) to any \( n \)?
3. How can the formula for \( a_n \) be used in more complex sequences or series?
4. What happens to the sum if we change the sequence to \( a_n = n^2 - 2n + 2 \)?
5. How do summation formulas apply when we are summing a quadratic function of \( n \)?

### Tip:
When summing sequences, it's often helpful to break down the terms one by one before adding them, especially if the expression involves polynomials.

Let a subscript n equals n squared minus 3 n plus 3, n greater or equal than 1. Find stack sum a subscript i with i equals 1 below and 4 on top.

Learn how to calculate the sum of the sequence n^2 - 3n + 3 for i running from 1 to 4. This problem involves step-by-step evaluation of the polynomial sequence and summing the results. Suitable for students in Grades 9-12 studying sequences and summation.

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