The sum of the first 4 terms of an arithmetic series is -8 and the sum of the first 5 terms is 85. Determine the first term and the common difference
So Sum(5) - Sum(4) = term(5)
85-(-8) = a + 4d
a+4d = 93 < (#1)
Sum(4) = -8 , since there are only 4 terms I will not use the sum formula
a + a+d + a+2d + a+3d = -8
4a + 6d = -8
2a + 3d = -4 , (#2)
2(#1) - (#2) ---> 5d = 190
d = 38
in #1, a = -59
check:
-59 - 21 + 17+ 55 = -8
sum(5) = sum(4) + term(5) = -8 + (-59+4(38)) = 85
To determine the first term and the common difference of an arithmetic series, we can use the formula for the sum of an arithmetic series:
S = (n/2)(2a + (n-1)d)
where:
S = sum of the terms in the series
n = number of terms in the series
a = first term of the series
d = common difference between the terms
Given that the sum of the first 4 terms is -8, we can substitute the values into the formula and solve for a:
-8 = (4/2)(2a + (4-1)d)
-8 = 2(2a + 3d)
-8 = 4a + 6d (equation 1)
Similarly, given that the sum of the first 5 terms is 85, we can substitute the values into the formula and solve for a:
85 = (5/2)(2a + (5-1)d)
85 = (5/2)(2a + 4d)
85 = 5a + 10d (equation 2)
Now we have a system of two equations (equations 1 and 2) with two unknowns (a and d). We can solve this system of equations to find the values of a and d.
First, let's multiply equation 1 by 5 and equation 2 by 2 to eliminate the fractional coefficients:
-40 = 20a + 30d (equation 3)
170 = 10a + 20d (equation 4)
Next, let's multiply equation 4 by 2 and subtract equation 3 from it:
(2 * 170) - (-40) = (2 * (10a + 20d)) - (20a + 30d)
340 + 40 = 20a + 40d - 20a - 30d
380 = 10d
Divide both sides of the equation by 10:
d = 38
Now substitute the value of d back into equation 3 to solve for a:
-40 = 20a + 30(38)
-40 = 20a + 1140
20a = -1180
a = -59
Therefore, the first term of the arithmetic series is -59 and the common difference is 38.