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.