sequences and series - Solve the arithmetic progression given the sum of first 4 elements, and the sum of the first and 5th element

I'm trying to solve an arithmetic progression (find the first few elements: a1, a2, a3, ...).

I'm given S4 (sum of the first 4 elements of a sequence) = 14 and the sum of the first a1 and 2 * 5th element, 2*a5 = 0.

Simply put: S4 = 14, a1 + 2a5=0

I tried using the formula (plugging in the values) for the sum of the first n-terms of an arithmetic sequence, but I can't solve it since I don't have neither a1 nor d.

What's the correct way of solving this?

Answer

$$S_4 = 14 \implies 2a + 3d = 7 \quad\quad \text{(1.) } $$

$$a+ 2a_5 = 0 \implies a + 2a + 8d = 0\implies a = -\frac{8d}{3} \quad\quad \text{(2.) }$$

Substituting $\text{(2.) }$ in $\text{(1.) }$

$$-\frac{16d}{3} + 3d = 7 \implies -\frac {7d}3 = 7 \implies d =-3 \quad\quad\text{(3.) }$$

From $\text{(2.) }$ and $\text{(3.) }$
$$a = -\frac83 . -3 \implies a=8$$

Hence the A.P is $ \,\, 8,5,2,-1,-4 ....$