To find the full arithmetic sequence, we can follow these steps:

Step 1: Identify the first term ($a$) and the common difference ($d$).

We are given the first three terms:

• First term, $a_1 = 28$
• Second term, $a_2 = 52$
• Third term, $a_3 = 82$

The common difference, $d$, in an arithmetic sequence is calculated by subtracting the first term from the second term: $d = a_2 - a_1 = 52 - 28 = 24$

Step 2: General formula for the $n$-th term of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ Substitute $a_1 = 28$ and $d = 24$ into the formula: $a_n = 28 + (n-1) \times 24$ $a_n = 28 + 24n - 24$ $a_n = 24n + 4$

Step 3: Use the sum formula for an arithmetic sequence

The sum of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (a_1 + a_n)$ We are told that the total sum of all terms is 1769: $S_n = 1769$

We can substitute the formula for $a_n$ and simplify the equation to find $n$.

The sum formula also can be expressed as: $S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$ Substitute $a_1 = 28$, $d = 24$, and $S_n = 1769$: $1769 = \frac{n}{2} \times (2 \times 28 + (n-1) \times 24)$ Simplify: $1769 = \frac{n}{2} \times (56 + 24n - 24)$ $1769 = \frac{n}{2} \times (24n + 32)$ $1769 = \frac{n}{2} \times (24n + 32)$ Multiply both sides by 2 to eliminate the fraction: $3538 = n \times (24n + 32)$ Expand the right side: $3538 = 24n^2 + 32n$ Rearrange the equation to form a quadratic: $24n^2 + 32n - 3538 = 0$

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For $24n^2 + 32n - 3538 = 0$, the coefficients are:

• $a = 24$
• $b = 32$
• $c = -3538$

Substitute these into the quadratic formula: $n = \frac{-32 \pm \sqrt{32^2 - 4 \times 24 \times (-3538)}}{2 \times 24}$ $n = \frac{-32 \pm \sqrt{1024 + 339648}}{48}$ $n = \frac{-32 \pm \sqrt{340672}}{48}$ $n = \frac{-32 \pm 583.56}{48}$

Step 5: Solve for $n$

We have two possible values for $n$: $n = \frac{-32 + 583.56}{48} \quad \text{or} \quad n = \frac{-32 - 583.56}{48}$ The negative solution doesn't make sense in this context, so we take the positive solution: $n = \frac{551.56}{48} \approx 11.49$ Since $n$ must be a whole number, we round to $n = 12$.

Step 6: Verify and find the sequence

Now that we know there are 12 terms, we can use the formula $a_n = 24n + 4$ to find all the terms in the sequence: $a_1 = 28, \quad a_2 = 52, \quad a_3 = 82, \quad \dots, \quad a_{12} = 24 \times 12 + 4 = 292$

The arithmetic sequence is: $28, 52, 82, 106, 130, 154, 178, 202, 226, 250, 274, 292$

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How do you find the sum of the first $n$ terms of an arithmetic sequence?
2. Can you derive the formula for the $n$-th term of an arithmetic sequence?
3. How do you solve a quadratic equation using the quadratic formula?
4. What is the relationship between the common difference and the terms in an arithmetic sequence?
5. How would the sequence change if the sum was different but the same common difference?

Tip: Always double-check your calculations, especially when using the quadratic formula, to avoid errors.