You should use loc to select by label:

FIRST = 3
LAST = 8

df_selected = df[df['fruit'] == 'pear']
out = df.loc[FIRST:LAST]

Or:

idx = df[df['fruit'] == 'pear'].index
out = df.loc[idx.min():idx.max()]

NB. since loc includes both ends, you do not need the +1.

Output:

   number   fruit  color letter
3       4    pear    red      B
4       5    pear  green      A
5       6    pear   blue      B
6       7  banana    red      A
7       8  banana  green      B
8       9    pear   blue      A

When using df_selected[1:2], this behaves like iloc and selects from the fourth to the ninth position (so just the row with label 8).

alternative

If your goal is to select all values between the first and last match, you could also use boolean indexing:

m = df['fruit'].eq('pear')
out = df[m.cummax()&m[::-1].cummax()]

How it works:

   number   fruit  color letter      m  m.cummax()  m[::-1].cummax()      &
0       1   apple    red      A  False       False              True  False
1       2   apple  green      B  False       False              True  False
2       3   apple   blue      A  False       False              True  False
3       4    pear    red      B   True        True              True   True
4       5    pear  green      A   True        True              True   True
5       6    pear   blue      B   True        True              True   True
6       7  banana    red      A  False        True              True   True
7       8  banana  green      B  False        True              True   True
8       9    pear   blue      A   True        True              True   True
9      10   apple    red      B  False        True             False  False