Let's analyze the given arithmetic sequence: $36, 38, 40, 42, \dots$.

1. Common Difference:

In an arithmetic sequence, the common difference ($d$) is the difference between consecutive terms. $d = 38 - 36 = 2$ Thus, the common difference is $d = 2$.

2. The 52nd Term:

The general formula for the $n$-th term ($a_n$) of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Where:

• $a_1 = 36$ (the first term),
• $d = 2$ (the common difference).

To find the 52nd term ($a_{52}$): $a_{52} = 36 + (52 - 1) \cdot 2 = 36 + 51 \cdot 2 = 36 + 102 = 138$ So, the 52nd term is $138$.

3. Explicit Formula:

The explicit formula for the $n$-th term of the sequence is: $a_n = 36 + (n - 1) \cdot 2$ Simplifying this: $a_n = 36 + 2n - 2 = 2n + 34$ Thus, the explicit formula is: $a_n = 2n + 34$

4. Recursive Formula:

The recursive formula for an arithmetic sequence expresses each term in terms of the previous one. The recursive formula is: $a_1 = 36, \quad a_n = a_{n-1} + 2 \quad \text{for} \, n \geq 2$

5. Three Terms After the Last Given:

The last given term is 42. To find the next three terms, we add the common difference ($d = 2$) to each successive term:

• First term: $42 + 2 = 44$
• Second term: $44 + 2 = 46$
• Third term: $46 + 2 = 48$

So, the three terms after 42 are: $44, 46, 48$.

6. Sum of the First 10 Terms:

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$ For the first 10 terms ($n = 10$): $a_1 = 36, \quad a_{10} = 2(10) + 34 = 54$ $S_{10} = \frac{10}{2} \cdot (36 + 54) = 5 \cdot 90 = 450$ Thus, the sum of the first 10 terms is $450$.

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Would you like further details or have any questions?

Here are five related questions:

1. How would you find the 100th term of the sequence?
2. What is the sum of the first 20 terms of this sequence?
3. How does the explicit formula change if the common difference is negative?
4. Can you derive the recursive formula from the explicit one?
5. What is the general formula for the sum of an arithmetic series?

Tip: The recursive formula is a quick way to find the next term if you know the previous one, but the explicit formula is faster for distant terms like the 52nd!