Precalculus Word Problem: setting up linear equation

Question

This is a math question, I got $18 I wonder if anyone can get the same answer and explain how.

Beatrice went shopping with #120. She bought items in five stores. In each store she spent $3 more than the previous. She went home with $25. How much did she spend in the first store?

Answer 1

Let $x$ be how much she spent in the first store: $$ \begin{align} (x) + (x+3) + (x+6) + (x+9) + (x+12) + 25 &= 120 \\ 5x + 55 &= 120 \\ 5x &= 65 \\ x &= 13 \end{align} $$

Answer 2

Another method :

Let $\left(u_n\right)_{n\geq1} \in \mathbb{N}^\mathbb{N}$ be a sequence where $u_n$ is the money she spent in the $n$th store.

$$u_{n+1}=u_n+3^\implies\forall n \in \mathbb{N}, u_n=u_0+3n$$ $$\sum_{k=0}^n u_k=(n+1)\frac{2u_0+3n}{2}$$ $$\implies u_0=\frac{2\sum_{k=0}^n u_k-3n(n+1)}{2(n+1)}$$

She comes homes with \$25 so $\sum_{k=0}^4 u_k = 120-25 = 95$.

Hence :

$$u_0 = \frac{2 \cdot 95-3 \cdot 4 \cdot 5}{2\cdot5}=13$$

So she spent \$13 in the first store.