We are given that the terms $3x + y$, $5x - 2y$, $7x - 5y$, ..., and the last term $21x - 35y$ form an arithmetic sequence. We need to determine $n$, which represents the number of terms in this sequence.

Step 1: General Form of Arithmetic Sequence

The general form of an arithmetic sequence is:

$a_1, a_2, a_3, \dots, a_n$

Where:

• $a_1$ is the first term.
• $a_n$ is the $n$-th term.
• The common difference between consecutive terms is constant, denoted by $d$.

Step 2: Calculate the Common Difference

We can compute the common difference $d$ using the first two terms:

1st term: $3x + y$
2nd term: $5x - 2y$

The common difference $d$ is:

$d = (5x - 2y) - (3x + y) = 5x - 2y - 3x - y = 2x - 3y$

So, the common difference is $2x - 3y$.

Step 3: General Term of Arithmetic Sequence

The general term of an arithmetic sequence is given by:

$a_n = a_1 + (n-1)d$

Here, $a_1 = 3x + y$ and $d = 2x - 3y$. Substituting these into the general formula:

$a_n = (3x + y) + (n-1)(2x - 3y)$

Simplifying:

$a_n = 3x + y + (n-1)(2x - 3y)$

$a_n = 3x + y + (2x(n-1) - 3y(n-1))$

$a_n = 3x + y + 2x(n-1) - 3y(n-1)$

$a_n = 3x + y + 2xn - 2x - 3yn + 3y$

$a_n = (3x - 2x) + 2xn + (y + 3y) - 3yn$

$a_n = x(1 + 2n) + y(4 - 3n)$

Step 4: Use the Last Term to Find $n$

The last term is given as $21x - 35y$. So, we equate it to the general term:

$x(1 + 2n) + y(4 - 3n) = 21x - 35y$

Now, equate the coefficients of $x$ and $y$:

For $x$:

$1 + 2n = 21$

Solving for $n$:

$2n = 20$

$n = 10$

Thus, the number of terms $n$ in the sequence is 10.

Final Answer:

The value of $n$ is 10.

Would you like more details or have any questions?

Here are 5 questions related to arithmetic sequences:

1. How do you find the common difference in any arithmetic sequence?
2. What is the general term formula for an arithmetic sequence?
3. How do you solve for $n$ when the last term of an arithmetic sequence is known?
4. What happens if the common difference in an arithmetic sequence is negative?
5. Can an arithmetic sequence have a fractional common difference?

Tip: Always check if your sequence fits the arithmetic progression rule by calculating the common difference between consecutive terms!