To determine the number of arithmetic progressions (A.P.s) with 10 terms, whose first term is in the set $\{1, 2, 3, 4\}$ and common difference is in the set $\{1, 2, 3, 4\}$, let’s break it down systematically:

Key Elements of an A.P.

1. The first term, $a$, is one of $\{1, 2, 3, 4\}$.
2. The common difference, $d$, is one of $\{1, 2, 3, 4\}$.
3. The A.P. with 10 terms is uniquely defined by $a$ and $d$, since the terms of the A.P. are given by: $a, a+d, a+2d, \ldots, a+9d.$

Counting the Number of A.P.s

• There are 4 choices for $a$ (from $\{1, 2, 3, 4\}$).
• There are 4 choices for $d$ (from $\{1, 2, 3, 4\}$).

Since the choices of $a$ and $d$ are independent, the total number of A.P.s is: $4 \times 4 = 16.$

Final Answer:

There are 16 arithmetic progressions.

Would you like detailed examples of specific A.P.s or further breakdowns of the concept?

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Relative Questions:

1. What defines an arithmetic progression, and how is it structured?
2. Can A.P.s have negative or fractional common differences?
3. How does the number of terms in an A.P. affect its properties?
4. What happens when the set of $a$ or $d$ changes in size?
5. How can A.P.s be used to solve real-world problems?

Tip:

To verify the correctness of an A.P., always check that the difference between consecutive terms remains constant!