Consider the arithmetic series 4-2-8.....-260.
Calculate the sum of the series
Determine m if 4-2-8....(to m terms) =-1700
Calculate the general value of k for which Sk>-800
To calculate the sum of the arithmetic series, we can use the formula Sn = n/2 * (a + L), where Sn is the sum of the series, n is the number of terms, a is the first term, and L is the last term.
In this case, the first term (a) is 4, the last term (L) is -260, and the common difference (d) can be found by subtracting the second term from the first term: d = -2 - 4 = -6.
To find the number of terms (n), we can use the formula L = a + (n-1)d: -260 = 4 + (n-1)(-6).
Simplifying this equation, we get -260 = 4 - 6n + 6.
Rearranging terms, we have -260 - 4 + 6 = -6n.
Simplifying further gives -260 + 2 = -6n, or -258 = -6n.
Dividing both sides by -6, we get n = 43.
Plugging the values of a, L, and n into the formula Sn = n/2 * (a + L), we find:
Sn = 43/2 * (4 + (-260))
= 21.5 * (-256)
= -5504
Therefore, the sum of the arithmetic series is -5504.
To find m if the sum of m terms is -1700, we can use the formula Sn = n/2 * (a + L) and solve for m:
-1700 = m/2 * (4 + (-260))
Dividing both sides by -278, we have:
-1700 / -278 = m/2
Simplifying gives:
6.115 = m/2
Multiplying both sides by 2, we find:
12.228 = m
Therefore, m is approximately 12.228.
To find the general value of k for which Sk > -800, we can use the formula Sk = n/2 * (2a + (n-1)d) and solve for k:
-800 = k/2 * (2*4 + (k-1)(-6))
Simplifying this equation gives:
-800 = k/2 * (8 - 6k + 6)
Expanding and rearranging terms, we have:
-800 = 4k - 3k^2 + 3k
This equation can be rewritten as a quadratic equation:
-3k^2 + 7k - 800 = 0
Solving this quadratic equation, we find two possible values for k:
k ≈ 17.723 or k ≈ -4.524
Therefore, the general value of k for which Sk > -800 is k > 17.723 or k < -4.524.