Determine the 10th and 21st terms of each of the arithmetic sequences: 4+7x; 5+9x; 6+ 11x.......
a21 = a1 + 20 d
a10 = a1 + 9 d
To find the 10th and 21st terms of each arithmetic sequence, we need to know the formula for the nth term of an arithmetic sequence.
The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n-1)d
Where:
an is the nth term of the sequence
a1 is the first term of the sequence
d is the common difference between terms
n is the position of the term in the sequence
Let's determine the 10th and 21st terms for each of the given arithmetic sequences.
1. For the sequence 4 + 7x:
In this sequence, the first term (a1) is 4 and the common difference (d) is 7.
To find the 10th term (a10), substitute the values into the formula:
a10 = a1 + (n-1)d
a10 = 4 + (10-1)*7
a10 = 4 + 63
a10 = 67
To find the 21st term (a21), substitute the values into the formula:
a21 = a1 + (n-1)d
a21 = 4 + (21-1)*7
a21 = 4 + 140
a21 = 144
So, the 10th term (a10) of the sequence 4 + 7x is 67 and the 21st term (a21) is 144.
2. For the sequence 5 + 9x:
In this sequence, the first term (a1) is 5 and the common difference (d) is 9.
To find the 10th term (a10), substitute the values into the formula:
a10 = a1 + (n-1)d
a10 = 5 + (10-1)*9
a10 = 5 + 81
a10 = 86
To find the 21st term (a21), substitute the values into the formula:
a21 = a1 + (n-1)d
a21 = 5 + (21-1)*9
a21 = 5 + 180
a21 = 185
So, the 10th term (a10) of the sequence 5 + 9x is 86 and the 21st term (a21) is 185.
3. For the sequence 6 + 11x:
In this sequence, the first term (a1) is 6 and the common difference (d) is 11.
To find the 10th term (a10), substitute the values into the formula:
a10 = a1 + (n-1)d
a10 = 6 + (10-1)*11
a10 = 6 + 99
a10 = 105
To find the 21st term (a21), substitute the values into the formula:
a21 = a1 + (n-1)d
a21 = 6 + (21-1)*11
a21 = 6 + 220
a21 = 226
So, the 10th term (a10) of the sequence 6 + 11x is 105 and the 21st term (a21) is 226.
Therefore, the 10th and 21st terms of each arithmetic sequence are as follows:
- Sequence 4 + 7x: 10th term = 67, 21st term = 144.
- Sequence 5 + 9x: 10th term = 86, 21st term = 185.
- Sequence 6 + 11x: 10th term = 105, 21st term = 226.
To find the 10th and 21st terms of the arithmetic sequences, we need to determine the general formula for the sequence, which is given by the formula:
\[a_n = a_1 + (n-1)d\]
where:
- \(a_n\) is the nth term of the sequence,
- \(a_1\) is the first term of the sequence, and
- \(d\) is the common difference between each term.
In this case, the first term, \(a_1\), varies with each sequence, while the common difference, \(d\), follows a pattern.
For the given sequences, we can observe the following patterns:
- For the sequence 4+7x:
- \(a_1\) = 4
- \(d\) = 3x (Since the difference between each term is 3x)
- For the sequence 5+9x:
- \(a_1\) = 5
- \(d\) = 4x (Since the difference between each term is 4x)
- For the sequence 6+11x:
- \(a_1\) = 6
- \(d\) = 5x (Since the difference between each term is 5x)
Now that we have determined the values of \(a_1\) and \(d\) for each sequence, we can find the 10th and 21st terms using the general formula.
For the 10th term:
- Plug in \(n = 10\) into the general formula:
- For the sequence 4+7x: \(a_{10} = 4 + (10-1)(3x)\)
- For the sequence 5+9x: \(a_{10} = 5 + (10-1)(4x)\)
- For the sequence 6+11x: \(a_{10} = 6 + (10-1)(5x)\)
For the 21st term:
- Plug in \(n = 21\) into the general formula:
- For the sequence 4+7x: \(a_{21} = 4 + (21-1)(3x)\)
- For the sequence 5+9x: \(a_{21} = 5 + (21-1)(4x)\)
- For the sequence 6+11x: \(a_{21} = 6 + (21-1)(5x)\)
Evaluate the expressions for each sequence to find the specific values of the 10th and 21st terms.