Let’s go through the questions step by step. The worksheet is about Arithmetic Sequences (A.S.).

Question 1:

You need to write the first 5 terms of the given arithmetic sequences (A.S.) based on the general terms provided.

(i) $T(n) = 4 + 5n$
To find the first 5 terms, substitute $n = 1, 2, 3, 4, 5$ into the formula:

• $T(1) = 4 + 5(1) = 9$
• $T(2) = 4 + 5(2) = 14$
• $T(3) = 4 + 5(3) = 19$
• $T(4) = 4 + 5(4) = 24$
• $T(5) = 4 + 5(5) = 29$

So, the first 5 terms are: 9, 14, 19, 24, 29.

(ii) $T(n) = 2n - 3$
Similarly, substitute $n = 1, 2, 3, 4, 5$:

• $T(1) = 2(1) - 3 = -1$
• $T(2) = 2(2) - 3 = 1$
• $T(3) = 2(3) - 3 = 3$
• $T(4) = 2(4) - 3 = 5$
• $T(5) = 2(5) - 3 = 7$

The first 5 terms are: -1, 1, 3, 5, 7.

Question 2:

You are asked to find the common difference and general term of the sequences.

(i) Sequence: 5, 12, 19, 26

• The common difference $d$ is the difference between consecutive terms:
  $12 - 5 = 7$, $19 - 12 = 7$, $26 - 19 = 7$, so $d = 7$.

• The general term of an arithmetic sequence is given by:
  $T(n) = a + (n-1) \cdot d$, where $a$ is the first term, and $d$ is the common difference.
  Here, $a = 5$, and $d = 7$, so:
  $T(n) = 5 + (n-1) \cdot 7 = 5 + 7n - 7 = 7n - 2$.
  Therefore, the general term is $T(n) = 7n - 2$.

(ii) Sequence: 3, -6, -15, -24

• The common difference $d$ is:
  $-6 - 3 = -9$, $-15 - (-6) = -9$, $-24 - (-15) = -9$, so $d = -9$.

• The first term $a = 3$, and using the formula for the general term:
  $T(n) = 3 + (n-1)(-9) = 3 - 9(n-1) = 3 - 9n + 9 = -9n + 12$.
  The general term is $T(n) = -9n + 12$.

Question 3:

You need to find the number of terms in the arithmetic sequence: 95, 87, 79, 71, ..., -1.

• First, find the common difference $d = 87 - 95 = -8$.
• The first term $a = 95$.
• The last term $l = -1$.
• The formula for the $n$-th term of an arithmetic sequence is:
  $T_n = a + (n-1) \cdot d$.

Setting $T_n = -1$, substitute $a = 95$ and $d = -8$:

$-1 = 95 + (n-1)(-8)$

Simplifying the equation:

$-1 = 95 - 8(n-1)$ $-1 = 95 - 8n + 8$ $-1 = 103 - 8n$ $8n = 103 + 1 = 104$ $n = \frac{104}{8} = 13$

So, the number of terms in the sequence is 13.

Let me know if you'd like more details or explanations on any part! Here are some follow-up questions for practice:

1. What would the 10th term of the sequence $T(n) = 4 + 5n$ be?
2. For the sequence $T(n) = 2n - 3$, what is the 6th term?
3. How would the general term change if the common difference was halved in sequence (i) of question 2?
4. In question 3, if the common difference was -6 instead of -8, how many terms would the sequence have?
5. Can you generalize how to find the number of terms when a sequence starts with any value and has any common difference?

Tip: When solving for the number of terms in an arithmetic sequence, always rearrange the general term formula properly before solving for $n$.