Consider the arithmetic sequence 6;10;4;...
1. Calculate the value of the 25th term in this sequence.
2. Calculate the sum of the first 25 terms of this sequence
3. How many terms of the sequence will add up to 1248
To find the values in this arithmetic sequence, we can first determine the common difference and then use the formula for the nth term of an arithmetic sequence.
1. First, let's find the common difference:
The common difference (d) can be found by subtracting any two consecutive terms.
The difference between the second term (10) and the first term (6) is 10 - 6 = 4.
Thus, the common difference in this sequence is 4.
2. To find the 25th term (aₙ), we use the formula:
aₙ = a₁ + (n - 1)d,
where aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference.
Plugging in the values we know:
a₁ = 6 (first term)
n = 25 (term number)
d = 4 (common difference)
aₙ = 6 + (25 - 1)4
aₙ = 6 + 24*4
aₙ = 6 + 96
aₙ = 102
Therefore, the value of the 25th term in this sequence is 102.
3. To find the sum of the first 25 terms (Sₙ), we use the formula:
Sₙ = (n/2)(2a₁ + (n - 1)d),
where Sₙ is the sum of the first n terms.
Plugging in the values we know:
n = 25 (25 terms)
a₁ = 6 (first term)
d = 4 (common difference)
Sₙ = (25/2)(2(6) + (25 - 1)4)
Sₙ = (25/2)(12 + 24*4)
Sₙ = (25/2)(12 + 96)
Sₙ = (25/2)(108)
Sₙ = 1350
Therefore, the sum of the first 25 terms of this sequence is 1350.
4. To find how many terms of the sequence will add up to 1248, we can set up an equation using the formula for the sum of an arithmetic sequence:
1248 = (n/2)(2a₁ + (n - 1)d)
Since we are solving for n, we rearrange the equation:
(n/2)(2a₁ + (n - 1)d) = 1248
Plugging in the values we know:
a₁ = 6 (first term)
d = 4 (common difference)
(n/2)(2(6) + (n - 1)4) = 1248
(n/2)(12 + 4n - 4) = 1248
(n/2)(4n + 8) = 1248
2n^2 + 4n = 1248
2n^2 + 4n - 1248 = 0
To solve this quadratic equation, we can factor it. However, it is simpler to divide the equation by 2 to simplify the coefficients:
n^2 + 2n - 624 = 0
Now we can attempt to factor this equation:
(n + 26)(n - 24) = 0
Setting each factor equal to zero:
n + 26 = 0 or n - 24 = 0
Solving for n:
n = -26 or n = 24
Since the number of terms cannot be negative, we discard n = -26.
Therefore, there are 24 terms in the sequence that will add up to 1248.