The nth term of a linear sequence

(a) $66$

1. $$7\times 10-4$$
2. Substitute n = 10 into the position rule.
3. 7 $\times$ 10 $-$ 4 = 66.

• P1 Establishing $7\times 10-4$ or an equivalent valid method.
• A1 Correct answer: $66$

(a) $4n + 3$

1. The terms go up by 4, so the $n$th term starts with $4n$.
2. When $n = 1$, $4n = 4$, but the first term is 7, so add 3: $4n + 3$.

• M1 Writing $4n$ as part of the expression.
• A1 The correct answer, $4n + 3$.

(b) $83$

1. $4 \times 20 + 3 = 83$.

• B1 The correct answer, $83$.

(a) $50$

1. $$6n+3=303$$
2. The common difference is 6, so the nth term is 6n + 3.
3. Solve 6n + 3 = 303: 6n = 300, so n = 50.

• P1 Establishing $6n+3=303$ or an equivalent valid method.
• A1 Correct answer: $50$

(a) $4n-1$

1. Use the common difference as the coefficient and adjust to match term 1.
   $$3-4=-1$$
2. Therefore $4n-1$.

• M1 Use the common difference as the coefficient and adjust to match term 1.
• A1 Correct answer: $4n-1$

(b) No. Solving gives $n=17+1/4$, which is not an integer.

1. Set the nth-term expression equal to the proposed value.
   $$4n-1=68$$
2. No. Solving gives $n=17+1/4$, which is not an integer.

• M1 Set the nth-term expression equal to the proposed value.
• C1 Correct conclusion with supporting reasoning: No. Solving gives $n=17+1/4$, which is not an integer.

(a) $3n+1$

1. Use the common difference and adjust to the first term.
   $$4-3=1$$
2. Therefore $3n+1$.

• M1 Use the common difference and adjust to the first term.
• A1 Correct answer: $3n+1$

(b) $33$

1. Keep the number of matches at most 100.
   $$3n+1\le 100$$
2. Therefore $33$.

• P1 Keep the number of matches at most 100.
• A1 Correct answer: $33$

(a) $6$

1. There are five equal steps between positions 3 and 8.
   $$\frac{45-15}{5}$$
2. Therefore $6$.

• M1 There are five equal steps between positions 3 and 8.
• A1 Correct answer: $6$

(b) $17$

1. Recover the first term and express the general term.
   $$6n-3=99$$
2. Therefore $17$.

• M1 Recover the first term and express the general term.
• A1 Correct answer: $17$

(a) No: $5n - 2 = 97$ gives $n = 19.8$, which is not a whole number.

1. Set the $n$th term equal to 97.
   $$5n - 2 = 97 \Rightarrow 5n = 99 \Rightarrow n = 19.8$$
2. $n$ is not a whole number, so 97 is not a term. (The 19th term is 93 and the 20th is 98.)

• M1 Solving $5n - 2 = 97$, or finding the terms either side of 97 (93 and 98).
• C1 "No", because $n$ is not a whole number (or 97 lies between the 19th and 20th terms).

(b) $203$

1. Solve the inequality.
   $$5n - 2 > 200 \Rightarrow 5n > 202 \Rightarrow n > 40.4$$
2. The smallest whole number $n$ is 41: $5 \times 41 - 2 = 203$.

• P1 Forming $5n - 2 > 200$ (or trying values of $n$ near 40).
• P1 Choosing $n = 41$.
• A1 The correct answer, $203$.

(a) $29$

1. Set the $n$th terms equal.
   $$3n + 2 = 47 - 2n \Rightarrow 5n = 45 \Rightarrow n = 9$$
2. The 9th terms are $3 \times 9 + 2 = 29$ and $47 - 18 = 29$.

• M1 Forming $3n + 2 = 47 - 2n$.
• A1 The correct answer, $29$.

(b) $5, 17, 29$

1. A: $5, 8, 11, 14, 17, 20, 23, 26, 29, \ldots$ (add 3). C: $5, 9, 13, 17, 21, 25, 29, \ldots$ (add 4).
2. Both start at 5. Common terms repeat every $\text{LCM}(3, 4) = 12$: $5, 17, 29$.

• P1 Listing enough terms of both sequences to find a common term, or finding that 5 is in both.
• P1 Recognising that common terms go up by 12 (the LCM of 3 and 4).
• A1 The correct answer, $5, 17, 29$.