Determine the 10th and 21st terms of the arithmetic sequence- 4+7x, 5+9x, 6+11x
5+9x - (4+7x) = 1+2x
a = 4+7x
d = 1+2x
T10 = a+9d = 4+7x + 9(1+2x) = 13+25x
T21 = a+20d = 4+7x + 20(1+2x) = 24+47x
T10 = 4 + 7x +(10-1)1+2x
= 4 + 7x +9(1+2x)
= 4 + 7x +9+18x
= 13+25x
T21 = 4+7x +(21-1)1+2x
= 4+7x +20(1+2x)
= 4+7x +20+40x
= 24+47x
To find the nth term of an arithmetic sequence, we can use the formula:
nth term = first term + (n - 1) * common difference
In this case, the first term is -4, and the common difference is 2x. Let's find the 10th term:
10th term = -4 + (10 - 1) * 2x
= -4 + 9 * 2x
= -4 + 18x
So, the 10th term is -4 + 18x.
Now let's find the 21st term:
21st term = -4 + (21 - 1) * 2x
= -4 + 20 * 2x
= -4 + 40x
So, the 21st term is -4 + 40x.
Therefore, the 10th term is -4 + 18x, and the 21st term is -4 + 40x.
To find the 10th and 21st terms of an arithmetic sequence, we need to determine the formula for the nth term of the sequence first.
In an arithmetic sequence, each term is found by adding a constant difference, denoted as "d," to the previous term.
In this case, the first term is -4+7x, the second term is 5+9x, and the third term is 6+11x.
To find the common difference, we need to calculate the difference between any two consecutive terms. Let's calculate the difference between the second and first terms:
(5+9x) - (-4+7x) = 5+9x + 4 -7x = 9+2x
Similarly, let's calculate the difference between the third and second terms:
(6+11x) - (5+9x) = 6+11x - 5-9x = 1+2x
Since we obtained the same difference of 2x in both calculations, we can conclude that the common difference is 2x.
Now that we know the common difference, we can find the formula for the nth term of the sequence. The formula is given by:
Term(n) = First Term + (n-1) * Common Difference
In this case, the first term is -4+7x and the common difference is 2x.
So, the formula for the nth term of the sequence is:
Term(n) = (-4+7x) + (n-1) * (2x)
Now, we can find the 10th and 21st terms by substituting the respective values of n into the formula:
10th Term: Term(10) = (-4+7x) + (10-1)*(2x)
= (-4+7x) + 9*(2x)
= -4 + 7x + 18x
= 25x - 4
21st Term: Term(21) = (-4+7x) + (21-1)*(2x)
= (-4+7x) + 20* (2x)
= -4 + 7x + 40x
= 47x - 4
Therefore, the 10th term of the sequence is 25x - 4, and the 21st term is 47x - 4.