We are given an arithmetic progression (AP) with 23 terms, and the sum of the AP is 483. Let's determine what we can conclude based on this information.

Formula for the Sum of an Arithmetic Progression:

The sum $S$ of an AP is given by: $S = \frac{n}{2} \cdot (a + l)$ where:

• $n$ is the number of terms,
• $a$ is the first term,
• $l$ is the last term.

Given:

• $S = 483$,
• $n = 23$.

Substituting the values: $483 = \frac{23}{2} \cdot (a + l)$

Simplifying: $483 = 11.5 \cdot (a + l)$ $a + l = \frac{483}{11.5} = 42$

So, the sum of the first and the last term of the AP is 42.

Now, let's analyze each option:

• (A) Sum of the first term and the last term is 46.
  This is incorrect, because we just calculated that $a + l = 42$, not 46.

• (B) If the first term is 4, then the last term is definitely 38.
  If $a = 4$, then: $a + l = 42 \implies 4 + l = 42 \implies l = 42 - 4 = 38$ This is correct.

• (C) If the first term is 8, then the last term is definitely 13.
  If $a = 8$, then: $a + l = 42 \implies 8 + l = 42 \implies l = 42 - 8 = 34$ So, this statement is incorrect (the last term would be 34, not 13).

• (D) Insufficient information to conclude (A), (B), or (C).
  This is incorrect, as we have enough information to conclude that option (B) is true.

Conclusion:

The correct answer is (B).

Would you like any further explanation or details? Here are some related questions you might find interesting:

1. How do we find the common difference in an arithmetic progression?
2. How do we calculate the nth term of an arithmetic progression?
3. What if the sum of an arithmetic progression is given in terms of variables?
4. Can the common difference of an AP be negative?
5. How can we determine the sum of an infinite arithmetic progression?

Tip: Always remember to verify the relationships between the first and last terms in an arithmetic progression, especially when calculating sums!