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Let's break down the questions from the image and solve them step by step.

### (a) The first four terms of a sequence \( S_n \) are: 89, 83, 77, 71.

#### (i) Find an expression for \( S_n \), the \( n \)-th term of this sequence.
The terms are decreasing in a linear pattern, so this is an arithmetic sequence.

- First term \( a = 89 \)
- Common difference \( d = 83 - 89 = -6 \)

The formula for the \( n \)-th term of an arithmetic sequence is:
\[
S_n = a + (n - 1) \cdot d
\]

Substitute the values of \( a \) and \( d \):
\[
S_n = 89 + (n - 1)(-6)
\]
Simplifying:
\[
S_n = 89 - 6(n - 1)
\]
\[
S_n = 89 - 6n + 6
\]
\[
S_n = 95 - 6n
\]

#### (ii) Find the smallest value of \( n \) for which \( S_n < 0 \).
We set \( S_n < 0 \) and solve for \( n \):
\[
95 - 6n < 0
\]
\[
95 < 6n
\]
\[
n > \frac{95}{6}
\]
\[
n > 15.83
\]
Thus, the smallest integer value of \( n \) is \( n = 16 \).

---

### (b) The \( n \)-th term of a different sequence \( T_n \) is given by \( T_n = n^2 - 4n \).

#### (i) Find and simplify an expression for \( T_{n+1} - T_n \).
We first compute \( T_{n+1} \):
\[
T_{n+1} = (n+1)^2 - 4(n+1)
\]
Expand both terms:
\[
T_{n+1} = (n^2 + 2n + 1) - 4(n + 1)
\]
\[
T_{n+1} = n^2 + 2n + 1 - 4n - 4
\]
\[
T_{n+1} = n^2 - 2n - 3
\]

Now subtract \( T_n \) from \( T_{n+1} \):
\[
T_{n+1} - T_n = (n^2 - 2n - 3) - (n^2 - 4n)
\]
Simplifying:
\[
T_{n+1} - T_n = n^2 - 2n - 3 - n^2 + 4n
\]
\[
T_{n+1} - T_n = 2n - 3
\]

#### (ii) The difference between \( T_{p+1} \) and \( T_p \) is 75. Find the value of \( p \).
We know:
\[
T_{p+1} - T_p = 2p - 3 = 75
\]
Solve for \( p \):
\[
2p - 3 = 75
\]
\[
2p = 78
\]
\[
p = 39
\]

---

### Summary of Answers:
- (a)(i) \( S_n = 95 - 6n \)
- (a)(ii) \( n = 16 \)
- (b)(i) \( T_{n+1} - T_n = 2n - 3 \)
- (b)(ii) \( p = 39 \)

Would you like more details or explanations on any step? Here are some related questions you might want to explore:

1. How is the common difference of an arithmetic sequence determined?
2. What is the general formula for the sum of an arithmetic sequence?
3. How do we derive the formula for \( T_{n+1} \) in quadratic sequences?
4. How can we solve more complex inequalities involving sequences?
5. How are quadratic sequences different from arithmetic sequences?

**Tip:** Always start with identifying patterns, like common differences in arithmetic sequences or degrees of terms in polynomial sequences, to simplify solving problems effectively.

The first four terms of a sequence, S_n, are 89, 83, 77, 71. Find an expression for S_n, the nth term of this sequence. Find the smallest value of n for which S_n < 0. The nth term of a different sequence, T_n, is given by T_n = n^2 - 4n. Find and simplify an expression for T_{n+1} - T_n. The difference between T_{p+1} and T_p is 75. Find the value of p.

Explore how to find the nth term of an arithmetic sequence and the difference between terms in a quadratic sequence. This problem covers algebraic concepts such as the formula for the nth term in arithmetic sequences and deriving expressions for quadratic sequences. Ideal for students in Grades 9-10.

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