Answer :
Let's break down the problem step-by-step.
The given sequence is: 4, 6, 8, 10, 12, ...
1. Find the eighth term of the sequence.
The given sequence is an arithmetic sequence where each term increases by a common difference of 2 (i.e., 6 - 4 = 2).
The formula for the n-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[ a_n = a + (n - 1)d \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term (4 in this case),
- [tex]\( d \)[/tex] is the common difference (2 in this case),
- [tex]\( n \)[/tex] is the term number we want to find (8 in this case).
Substituting the values into the formula gives us:
[tex]\[ a_8 = 4 + (8 - 1) \times 2 \][/tex]
[tex]\[ a_8 = 4 + 7 \times 2 \][/tex]
[tex]\[ a_8 = 4 + 14 \][/tex]
[tex]\[ a_8 = 18 \][/tex]
So, the eighth term of the sequence is 18.
2. Find the sum of the first 25 terms of the sequence.
The sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n \)[/tex] of an arithmetic sequence can be calculated using the formula:
[tex]\[ S_n = \frac{n}{2} \times [2a + (n - 1)d] \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of terms (25 in this case).
Substituting the values into the formula gives us:
[tex]\[ S_{25} = \frac{25}{2} \times [2 \times 4 + (25 - 1) \times 2] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times [8 + 24 \times 2] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times [8 + 48] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times 56 \][/tex]
[tex]\[ S_{25} = \frac{25 \times 56}{2} \][/tex]
[tex]\[ S_{25} = \frac{1400}{2} \][/tex]
[tex]\[ S_{25} = 700 \][/tex]
So, the sum of the first 25 terms of the sequence is 700.
3. How many terms' sum is equal to the value of the 229th term?
Firstly, we find the value of the 229th term using the n-th term formula again:
[tex]\[ a_{229} = 4 + (229 - 1) \times 2 \][/tex]
[tex]\[ a_{229} = 4 + 228 \times 2 \][/tex]
[tex]\[ a_{229} = 4 + 456 \][/tex]
[tex]\[ a_{229} = 460 \][/tex]
Now, we need to find the number of terms [tex]\( n \)[/tex] such that the sum of the first [tex]\( n \)[/tex] terms equals 460.
Using the sum formula again:
[tex]\[ S_n = \frac{n}{2} \times [2a + (n - 1)d] = 460 \][/tex]
Substituting the known values into the equation:
[tex]\[ \frac{n}{2} \times [2 \times 4 + (n - 1) \times 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [8 + (n - 1) \times 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [8 + 2n - 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [2n + 6] = 460 \][/tex]
[tex]\[ n \times (n + 3) = 460 \][/tex]
By solving the quadratic equation [tex]\( n \times (n + 3) = 460 \)[/tex], we find:
[tex]\[ n = -23 \][/tex]
(The negative value does not make sense in this context. The equation likely has no real, positive solution.)
So, the number of terms' sum that equals the value of the 229th term doesn't exist in the sequence for a positive integer [tex]\( n \)[/tex]. Therefore, the correct term count which fits the described criteria is -23 (though this is a technical anomaly).
The given sequence is: 4, 6, 8, 10, 12, ...
1. Find the eighth term of the sequence.
The given sequence is an arithmetic sequence where each term increases by a common difference of 2 (i.e., 6 - 4 = 2).
The formula for the n-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[ a_n = a + (n - 1)d \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term (4 in this case),
- [tex]\( d \)[/tex] is the common difference (2 in this case),
- [tex]\( n \)[/tex] is the term number we want to find (8 in this case).
Substituting the values into the formula gives us:
[tex]\[ a_8 = 4 + (8 - 1) \times 2 \][/tex]
[tex]\[ a_8 = 4 + 7 \times 2 \][/tex]
[tex]\[ a_8 = 4 + 14 \][/tex]
[tex]\[ a_8 = 18 \][/tex]
So, the eighth term of the sequence is 18.
2. Find the sum of the first 25 terms of the sequence.
The sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n \)[/tex] of an arithmetic sequence can be calculated using the formula:
[tex]\[ S_n = \frac{n}{2} \times [2a + (n - 1)d] \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of terms (25 in this case).
Substituting the values into the formula gives us:
[tex]\[ S_{25} = \frac{25}{2} \times [2 \times 4 + (25 - 1) \times 2] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times [8 + 24 \times 2] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times [8 + 48] \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times 56 \][/tex]
[tex]\[ S_{25} = \frac{25 \times 56}{2} \][/tex]
[tex]\[ S_{25} = \frac{1400}{2} \][/tex]
[tex]\[ S_{25} = 700 \][/tex]
So, the sum of the first 25 terms of the sequence is 700.
3. How many terms' sum is equal to the value of the 229th term?
Firstly, we find the value of the 229th term using the n-th term formula again:
[tex]\[ a_{229} = 4 + (229 - 1) \times 2 \][/tex]
[tex]\[ a_{229} = 4 + 228 \times 2 \][/tex]
[tex]\[ a_{229} = 4 + 456 \][/tex]
[tex]\[ a_{229} = 460 \][/tex]
Now, we need to find the number of terms [tex]\( n \)[/tex] such that the sum of the first [tex]\( n \)[/tex] terms equals 460.
Using the sum formula again:
[tex]\[ S_n = \frac{n}{2} \times [2a + (n - 1)d] = 460 \][/tex]
Substituting the known values into the equation:
[tex]\[ \frac{n}{2} \times [2 \times 4 + (n - 1) \times 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [8 + (n - 1) \times 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [8 + 2n - 2] = 460 \][/tex]
[tex]\[ \frac{n}{2} \times [2n + 6] = 460 \][/tex]
[tex]\[ n \times (n + 3) = 460 \][/tex]
By solving the quadratic equation [tex]\( n \times (n + 3) = 460 \)[/tex], we find:
[tex]\[ n = -23 \][/tex]
(The negative value does not make sense in this context. The equation likely has no real, positive solution.)
So, the number of terms' sum that equals the value of the 229th term doesn't exist in the sequence for a positive integer [tex]\( n \)[/tex]. Therefore, the correct term count which fits the described criteria is -23 (though this is a technical anomaly).
