Let's solve these questions step-by-step:

Question 9:

The sequence is an arithmetic sequence: $\{5, 5+5\cdot1, 5+5\cdot2, 5+5\cdot3, \dots, 5+5\cdot19\cdot1\}$

This can be rewritten as: $a_n = 5 + (n-1)\cdot 5$ Where:

• $a_1 = 5$,
• $d = 5$, and
• The last term is $a_n = 5+19\cdot1$, which simplifies to $a_n = 24$.

Using the formula for the $n$-th term: $a_n = a_1 + (n-1)\cdot d$

Plugging $a_n = 24$, $a_1 = 5$, and $d = 5$: $24 = 5 + (n-1)\cdot 5$ $24 - 5 = (n-1)\cdot 5$ $19 = (n-1)\cdot 5$ $n-1 = 19 / 5$ $n = 5$

Answer: 5 terms

Question 10:

The statement describes the property of an arithmetic sequence (constant difference between consecutive terms). However, a geometric sequence is defined by the ratio between consecutive terms being constant, not the difference.

Answer: False

Question 11:

The general term of a geometric sequence is given by: $a_n = a_1 \cdot r^{n-1}$ Where:

• $a_1 = 10$,
• $r = 4$.

Substitute the given values: $a_n = 10 \cdot 4^{n-1}$

Answer: $a_n = 10 \cdot 4^{n-1}$

Would you like more details or clarification on any of the steps? Let me know!

5 Related Questions:

1. How do you distinguish between arithmetic and geometric sequences?
2. Can an arithmetic sequence have fractional terms? Provide an example.
3. What is the formula for the sum of the first $n$ terms in an arithmetic sequence?
4. How would you solve for the general term of a geometric sequence when the ratio is a fraction?
5. Can a sequence be both arithmetic and geometric? If so, under what conditions?

Tip:

For arithmetic sequences, focus on the difference $d$. For geometric sequences, look at the ratio $r$. Recognizing these patterns is key to solving sequence problems.