1. Find the 12th term of the arithmetic sequence 2, 6, 10, … .

2. Solve for the 101st term of the sequence whose 1st term is x-y and d=2x+y-3.

3. In the sequence 2, 6, 10, … , what term has a value of 106?

#1 and #2 are the same type as the ones I just did for you

Give them a try

#3
a=2 , d=4

a+(n-1)d = term(n)
2 + (n-1)(4) = 106
2 + 4n - 4 = 106
4n = 108
n = 27

term(27) = 106

Okay. I think I get it. Thanks for the help.

1. To find the 12th term of an arithmetic sequence, we need to determine the common difference and use the formula for the nth term.

In this sequence, the common difference is 4, as each term increases by 4 compared to the previous term.

The formula for the nth term of an arithmetic sequence is:
nth term = first term + (n - 1) * common difference.

Substituting the given values, we have:
12th term = 2 + (12 - 1) * 4
= 2 + 11 * 4
= 2 + 44
= 46.

Therefore, the 12th term of the arithmetic sequence 2, 6, 10, ... is 46.

2. To find the 101st term of a sequence with a given formula, we need to substitute the values into the formula for the nth term.

The formula for the nth term of the sequence is:
nth term = first term + (n - 1) * common difference.

Given that the first term is (x - y) and the common difference is (2x + y - 3), we can substitute those values into the formula:
101st term = (x - y) + (101 - 1) * (2x + y - 3)
= (x - y) + 100 * (2x + y - 3).

Therefore, the 101st term of the sequence is (x - y) + 100 * (2x + y - 3).

3. To determine which term in the sequence has a value of 106, we can use the formula for the nth term of an arithmetic sequence.

Given that the first term is 2, and the common difference is 4, we can set up the equation:
nth term = 2 + (n - 1) * 4.

To find the term with a value of 106, we substitute the value into the equation and solve for n:
106 = 2 + (n - 1) * 4.

Simplifying the equation, we have:
106 = 2 + 4n - 4,
104 = 4n - 2,
106 = 4n,
n = (106/4),
n = 26.5.

Since n cannot be a fraction, we can conclude that there is no term with a value of exactly 106 in this sequence.

1. To find the 12th term of the arithmetic sequence 2, 6, 10, ..., we need to determine the common difference (d) first. The common difference is the constant value added to each term to obtain the next term. To find the common difference, we can subtract any term from the one following it. In this case, we subtract 2 from 6 to get 4. This means the common difference (d) is 4.

Now that we have the common difference, we can use the formula for the nth term of an arithmetic sequence:

nth term = first term + (n - 1) * common difference

Substituting in the values we know, we have:

12th term = 2 + (12 - 1) * 4
= 2 + 11 * 4
= 2 + 44
= 46

Therefore, the 12th term of the arithmetic sequence 2, 6, 10, ..., is 46.

2. To solve for the 101st term of the given sequence with first term x-y and common difference d = 2x+y-3, we can use the same formula as mentioned earlier:

nth term = first term + (n - 1) * common difference

Substituting in the values we know, we have:

101st term = (x - y) + (101 - 1) * (2x + y - 3)
= (x - y) + 100 * (2x + y - 3)

This is the expression for the 101st term of the sequence.

3. To determine which term in the sequence 2, 6, 10, ..., has a value of 106, we can use the formula for the nth term of an arithmetic sequence and solve for n:

nth term = first term + (n - 1) * common difference

Since we know the first term (2) and the common difference (4), we can substitute these values in the formula:

106 = 2 + (n - 1) * 4

Simplifying, we have:

106 = 2 + 4n - 4
= 4n - 2

Adding 2 to both sides, we get:

108 = 4n

Dividing both sides by 4, we have:

27 = n

Therefore, the term with a value of 106 in the sequence 2, 6, 10, ..., is the 27th term.