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What is the common difference, denoted as $$d$$, in an arithmetic sequence?

In the formula $$T_n = a + (n - 1)d$$ for an arithmetic sequence, what does $$n$$ represent?

How do you determine if a sequence is arithmetic?

What does the graph of an arithmetic sequence look like when $$T_n$$ is plotted against $$n$$?

In a geometric sequence, what does the common ratio (r) represent?

How is the geometric mean calculated between two numbers $$a$$ and $$b$$?

What type of graph is produced when plotting $$T_n$$ against $$n$$ for a geometric sequence?

Given the arithmetic sequence 2, 5, 8, 11, ..., what is the common difference?

Find the 10th term of the arithmetic sequence where the first term is 3 and the common difference is 4.

Determine the arithmetic mean between 7 and 15.

If the first term of a geometric sequence is 2 and the common ratio is 3, what is the 4th term?

What is the geometric mean between 4 and 9?

Which condition must be met for an infinite geometric series to converge?

Which of the following is an example of a finite series?

Express the sum of the first 5 terms of a sequence $$T_i$$ using sigma notation.

What is the sum of the infinite geometric series with first term $$a = 4$$ and common ratio $$r = \frac{1}{2}$$?

What formula is used to calculate the sum of a finite arithmetic series when the first term $$(a)$$, last term $$(l)$$, and number of terms $$(n)$$ are known?

The sum of the first $$n$$ terms of an arithmetic series is given by $$S_n = \frac{n}{2}[2a + (n-1)d]$$. If $$a = 5$$, $$d = 3$$, and $$n = 10$$, find $$S_{10}$$.

What is the sum of the first 100 positive integers?

The first term of an arithmetic sequence is 2, and the common difference is 5. If the $$n$$-th term is 62, what is the value of $$n$$?

The fourth term of a geometric sequence is 24, and the common ratio is 2. What is the first term?

Find the sum of the first 6 terms of the geometric series: 3 + 6 + 12 + 24 + ...

The second term of an arithmetic sequence is 7, and the fourth term is 15. Find the common difference.

Determine the sum of the infinite geometric series: $$9 + 3 + 1 + \frac{1}{3} + \ldots$$

Suppose the sum of an infinite geometric series is 36 and the common ratio is $$\frac{1}{3}$$. What is the first term?

For an arithmetic sequence, the sum of the first 6 terms is 57, and the first term is 2. What is the common difference?

The second and fifth terms of a geometric sequence are 6 and -48, respectively. What is the common ratio?

If the sum of an infinite geometric series is 64, and the first term is 16, find the common ratio.

The arithmetic mean of two numbers is 13, and their geometric mean is 12. What are the two numbers?

Given an arithmetic series with first term $$a$$, common difference $$d$$, and last term $$l$$, derive a formula for the sum of the series $$S_n$$ in terms of $$a$$, $$d$$, and $$l$$ only, without explicitly using $$n$$.

Consider the infinite geometric series $$\sum_{n=1}^{\infty} ar^{n-1}$$. Given that the sum of the series is $$S$$ and the sum of the squares of the terms is also $$S$$, find the value of $$a$$ in terms of $$S$$. Assume $$a > 0$$.

Given a sequence where the term $$T_n = n^2 + 1$$, calculate the value of the sum $$\sum_{n=1}^{5} T_n$$.

What is the defining characteristic of an arithmetic sequence?

In the arithmetic sequence formula, $T_n = a + (n - 1)d$, which component represents the first term of the sequence?

If the first term of an arithmetic sequence is 4 and the common difference is -2, what is the third term?

To confirm if a sequence is arithmetic, what must be consistent between consecutive terms?

What is the arithmetic mean between two numbers, 10 and 20?

Graphically, what shape is formed when plotting the terms of an arithmetic sequence ($T_n$) against their position ($n$)?

If the differences between consecutive terms in a sequence are not constant, can the sequence be arithmetic?

What does a positive common difference in an arithmetic sequence indicate about the sequence?

Which of the following sequences is an arithmetic sequence?

What is the defining characteristic of a geometric sequence?

In the formula $T_n = ar^{n-1}$ for a geometric sequence, what does 'r' represent?

If the first term of a geometric sequence is 2 and the common ratio is 3, what is the second term?

To check if a sequence is geometric, what should be constant between consecutive terms?

What type of graph is formed when plotting terms of a geometric sequence ($T_n$) against their position ($n$)?

If the ratios between consecutive terms in a sequence are not constant, can the sequence be geometric?

What does a common ratio $r > 1$ in a geometric sequence indicate about the sequence's growth?

Which of the following sequences is a geometric sequence?

What is a series?

What distinguishes a finite series from an infinite series?

What does the symbol '$\Sigma$' represent in sigma notation?

In the sigma notation $\sum_{i=m}^{n} T_i$, what does 'i' represent?

What is the formula for the sum of the first $n$ terms of a finite geometric series?

Under what condition does an infinite geometric series converge?

What is the formula for the sum of a convergent infinite geometric series?

For a finite arithmetic series, what does 'l' represent in the formula $S_n = \frac{n}{2}(a + l)$?

What is the sum of the first 5 terms of the arithmetic series if the first term is 2 and the common difference is 3?

What method did Karl Friedrich Gauss famously use to calculate the sum of the first 100 positive integers?

For an arithmetic series, which formula is used when the last term ($l$) is known instead of the common difference ($d$)?

An infinite geometric series has a first term of 6 and a common ratio of $-\frac{1}{2}$. What is its sum?

For what values of the common ratio 'r' does a geometric sequence alternate in sign?

Consider a geometric sequence with first term $a$ and common ratio $r$. If the second term is 10 and the fourth term is 40, what is the common ratio $r$ (assuming $r>0$)?

The sum of the first two terms of an arithmetic sequence is 8, and the sum of the first four terms is 24. What is the first term?

The third term of a geometric sequence is 12 and the sixth term is 96. What is the common ratio $r$?

A sequence is defined by $T_n = 3n^2 - 2$. What is the sum of the first 3 terms?

The sum of an infinite geometric series is 27 and the first term is 18. What is the common ratio?

If the sum of the first $n$ terms of an arithmetic series is given by $S_n = 2n^2 + 3n$, what is the common difference of this series?

Consider two arithmetic sequences. The first sequence starts with 5 and has a common difference of 3. The second sequence starts with 2 and has a common difference of 4. For which term number $n$ will the $n$-th term of the second sequence first exceed the $n$-th term of the first sequence?