Finding the First Term and Common Difference of an Arithmetic Series

[Blister]

Homework Statement

The sum of the first 4 terms in an arithmetic series is -8 and the sum of the first 5 terms is 85. Determine the first term and the common difference.

Homework Equations

tn = a + (n-1)d
Sn = n/2 (2a + (n-1)d)

The Attempt at a Solution

 

[rock.freak667]

The Attempt at a Solution

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[Architeuthis Dux]

Using the given equations, we can set up a system of equations to solve for the first term (a) and the common difference (d).

For the first equation, we can substitute n=4 and Sn=-8:

-8 = 4/2 (2a + (4-1)d)
-8 = 2(2a+3d)
-8 = 4a+6d

For the second equation, we can substitute n=5 and Sn=85:

85 = 5/2 (2a + (5-1)d)
85 = 5(2a+4d)
85 = 10a+20d

We now have a system of equations with two unknowns (a and d). To solve this, we can use the method of elimination. By multiplying the first equation by 5 and the second equation by 2, we can eliminate the variable "a" and solve for "d". This gives us:

-40 = 20a+30d
170 = 20a+40d

Subtracting the two equations, we get:

-210 = 10d
d = -21

Now, we can substitute this value of d back into either of the original equations to solve for a. Using the first equation, we get:

-8 = 4a+6(-21)
-8 = 4a-126
118 = 4a
a = 29.5

Therefore, the first term of the series is 29.5 and the common difference is -21. We can check our solution by plugging these values into the equations:

t4 = 29.5 + (4-1)(-21) = -44.5
t5 = 29.5 + (5-1)(-21) = -65.5
S4 = 4/2 (2(29.5) + (4-1)(-21)) = -8
S5 = 5/2 (2(29.5) + (5-1)(-21)) = 85

Therefore, our solution is correct and the first term and common difference for the given arithmetic series are 29.5 and -21, respectively.