Output for a decaying stem-plot h[n]

From the figure, $h[n]=\{2,\ 1,\ 0.5,\ 0.25,\ 0.125\}$ for $n=0..4$, i.e. $h[n]=2(0.5)^n$ for $0\le n\le4$. The input is $x[n]=\{-0.25,\ 0.5,\ 1,\ -0.5,\ 0,\ 0.25\}$ with $x[0]=1$, so $x[-2]=-0.25,\ x[-1]=0.5,\ x[0]=1,\ x[1]=-0.5,\ x[2]=0,\ x[3]=0.25$.

The output $y[n]=\sum_k x[k]h[n-k]$ starts at $n=-2+0=-2$ and ends at $n=3+4=7$ (length $6+5-1=10$).

Tabular method

Each row is $x[k]\cdot h[n-k]$:

Some values worked out

$$\begin{aligned} y[0]&=x[-2]h[2]+x[-1]h[1]+x[0]h[0]=-0.125+0.5+2=2.375\\ y[1]&=x[-2]h[3]+x[-1]h[2]+x[0]h[1]+x[1]h[0]=-0.0625+0.25+1-1=0.1875\\ y[3]&=x[-1]h[4]+x[0]h[3]+x[1]h[2]+x[2]h[1]+x[3]h[0]=0.0625+0.25-0.25+0+0.5=0.5625 \end{aligned}$$

Sum check: $\sum y=3.875=(\sum x)(\sum h)=1\times3.875$.

Answer:

and $y[n]=0$ elsewhere.

Plot of y[n]

(Not to scale for the small values.)

    y[n]
  2.375           o
   0.75      o    |
 0.5625      |    |              o
   0.25      |    |              |    o
 0.1875      |    |    o         |    |
0.09375      |    |    |    o    |    |
 0.0625      |    |    |    |    |    |    o    o
0.03125      |    |    |    |    |    |    |    |    o
      0 +----+----+----+----+----+----+----+----+----+--> n
   -0.5 o
        -2   -1   0    1    2    3    4    5    6    7