What is the value of the 30th term in the following arithmetic sequence 12,6,0,6
To find the value of the 30th term in the arithmetic sequence with terms 12, 6, 0, 6, we need to identify the pattern and calculate the general formula for the sequence.
From the given sequence, we can observe that the common difference is -6, as each term is obtained by subtracting 6 from the previous term.
To find the value of the nth term in an arithmetic sequence, we can use the formula:
an = a1 + (n - 1)d
Where:
an is the value of the nth term
a1 is the first term
n is the term number
d is the common difference
In our case, the first term (a1) is 12, the term number (n) is 30, and the common difference (d) is -6.
Plugging the values into the formula, we can calculate the 30th term:
a30 = 12 + (30 - 1)(-6)
a30 = 12 + (29)(-6)
a30 = 12 - 174
a30 = -162
Therefore, the value of the 30th term in the arithmetic sequence 12, 6, 0, 6 is -162.
To find the value of the 30th term in an arithmetic sequence, we need to first determine the common difference (d).
In an arithmetic sequence, the common difference (d) is the constant amount added or subtracted to each term to obtain the next term.
Looking at the given sequence, we can observe that the common difference is changing. It is alternating between -6 and +6.
To find a pattern, we can rewrite the sequence as follows:
12, 6, 0, 6, ...
Here, we see that the terms are repeating every four terms. In the first four terms, the common difference is -6. In the next four terms, it is +6.
So, we can write the general form of this sequence as follows:
a + (n - 1) * d
Where:
a = first term = 12
n = term number = 30
d = common difference
Since the sequence has four term patterns, to find the exact common difference for the 30th term, we can use modular arithmetic (n modulus 4) to determine the position of the 30th term within the repeating pattern.
The remainder when 30 is divided by 4 is 2. Therefore, the 30th term will be in the second position of the repeating pattern.
For the second position, the common difference is +6.
Now, we can calculate the value of the 30th term by plugging in the values:
a + (n - 1) * d
= 12 + (30 - 1) * 6
= 12 + 29 * 6
= 12 + 174
= 186
Hence, the value of the 30th term in the given arithmetic sequence is 186.
I assume you meant 12,6,0,-6,...
If so, then clearly, a=6, d = -6
a30 = a + 29d
So plug and chug.