1st Question: Which of the following is not an Arithmetic Progression (AP)?

Let's analyze each sequence:

1. 1, 4, 7, 10,...

   • The common difference is $4 - 1 = 3$, $7 - 4 = 3$, and $10 - 7 = 3$.
   • This is an AP.

2. 3, 7, 12, 16,...

   • The common difference is $7 - 3 = 4$, $12 - 7 = 5$, and $16 - 12 = 4$.
   • The difference is not consistent, so this is not an AP.

3. -5, -2, 1, 4,...

   • The common difference is $-2 - (-5) = 3$, $1 - (-2) = 3$, and $4 - 1 = 3$.
   • This is an AP.

4. 11, 14, 17, 20,...

   • The common difference is $14 - 11 = 3$, $17 - 14 = 3$, and $20 - 17 = 3$.
   • This is an AP.

So, the answer is 3, 7, 12, 16 (Option 2).

2nd Question: Evaluate the geometric series $3 + 15 + 75 + 375 + \dots$ for $n = 6$.

This is a geometric progression with:

• First term $a = 3$
• Common ratio $r = \frac{15}{3} = 5$
• Number of terms $n = 6$

The sum $S_n$ of a geometric series is given by the formula: $S_n = a \frac{r^n - 1}{r - 1}$ Substitute the values: $S_6 = 3 \frac{5^6 - 1}{5 - 1} = 3 \frac{15625 - 1}{4} = 3 \frac{15624}{4} = 3 \times 3906 = 11718$

The answer is 11,718.

Would you like more details on any of these solutions or have any questions? Here are five related questions you could explore:

1. What defines an arithmetic progression?
2. How can you identify a geometric progression from a sequence of numbers?
3. What happens to the sum of a geometric series when the common ratio is less than 1?
4. How do you derive the formula for the sum of an arithmetic progression?
5. Can a sequence be neither arithmetic nor geometric? How would that look?

Tip: For geometric series, remember that the common ratio is the key factor; any change in it can significantly alter the sum calculation.