The sum of the first two terms of an arithmetic series is 15 and the sum of the next two terms is 43. What are the first four terms of the series?
a + a+d = 15
a+2d + a+3d = 43
collect terms to get
2a+d = 15
2a+5d = 43
That should get you started.
To find the first four terms of the arithmetic series, we need to find the common difference (d) and the first term (a₁).
Let's use the given information to set up equations.
Given:
The sum of the first two terms is 15.
The sum of the next two terms is 43.
Let's use the formula for the sum of terms in an arithmetic series to set up these equations:
The sum of the first two terms: S₂ = (n/2)(2a₁ + (n-1)d) = 15
The sum of the next two terms: S₄ = (n/2)(2a₁ + 3d) = 43
Since we want to find the first four terms, we can assume that n = 4.
Substituting n = 2 and n = 4 into the above equations:
1. S₂ = (2/2)(2a₁ + (2-1)d) = 15
2. S₄ = (4/2)(2a₁ + 3d) = 43
Simplifying the equations:
1. 2a₁ + d = 15
2. 2a₁ + 3d = 43
Now we have a system of two equations with two unknowns. We can solve these equations simultaneously to find the values of `a₁` and `d`. Once we find these values, we can determine the first four terms of the series.
Solving the equations:
Subtracting equation 1 from equation 2 to eliminate `a₁`:
(2a₁ + 3d) - (2a₁ + d) = 43 - 15
2d = 28
d = 14
Substituting the value of `d` into equation 1 to solve for `a₁`:
2a₁ + 14 = 15
2a₁ = 1
a₁ = 1/2
So, the common difference (`d`) is 14 and the first term (`a₁`) is 1/2.
Now, we can find the first four terms of the series:
a₁ = 1/2
a₂ = a₁ + d = 1/2 + 14 = 29/2
a₃ = a₂ + d = 29/2 + 14 = 43/2
a₄ = a₃ + d = 43/2 + 14 = 57/2
Therefore, the first four terms of the series are 1/2, 29/2, 43/2, and 57/2.