Find as indicated in each case:
(i) `t_(1)=1,t_(n)=2t_(n-1),(ngt 1),t_(6)=?`
(ii)`S_(n)=S_(n-1)-1,(ngt2), S_(1)=S_(2)=2,S_(5)=?`

Answer

Let's solve the given problems step by step.

### Part (i): Find \( t_6 \)

We are given:
- \( t_1 = 1 \)
- \( t_n = 2 \cdot t_{n-1} \) for \( n > 1 \)

We need to find \( t_6 \).

**Step 1:** Calculate \( t_2 \)
\[ t_2 = 2 \cdot t_1 = 2 \cdot 1 = 2 \]

**Step 2:** Calculate \( t_3 \)
\[ t_3 = 2 \cdot t_2 = 2 \cdot 2 = 4 \]

**Step 3:** Calculate \( t_4 \)
\[ t_4 = 2 \cdot t_3 = 2 \cdot 4 = 8 \]

**Step 4:** Calculate \( t_5 \)
\[ t_5 = 2 \cdot t_4 = 2 \cdot 8 = 16 \]

**Step 5:** Calculate \( t_6 \)
\[ t_6 = 2 \cdot t_5 = 2 \cdot 16 = 32 \]

Thus, the value of \( t_6 \) is \( 32 \).

### Part (ii): Find \( S_5 \)

We are given:
- \( S_n = S_{n-1} - 1 \) for \( n > 2 \)
- \( S_1 = 2 \)
- \( S_2 = 2 \)

We need to find \( S_5 \).

**Step 1:** Calculate \( S_3 \)
\[ S_3 = S_2 - 1 = 2 - 1 = 1 \]

**Step 2:** Calculate \( S_4 \)
\[ S_4 = S_3 - 1 = 1 - 1 = 0 \]

**Step 3:** Calculate \( S_5 \)
\[ S_5 = S_4 - 1 = 0 - 1 = -1 \]

Thus, the value of \( S_5 \) is \( -1 \).

### Summary of Results
- \( t_6 = 32 \)
- \( S_5 = -1 \)