m by 2, then subtract 1. Find the 5th term. 1. Start with the first term: 3. 2. Double 3 and subtract 1 to get the second term: 5. 3. Double 5 and subtract 1 to get the third term: 9. 4. Double 9 and subtract 1 to get the fourth term: 17. 5. Double 17 and subtract 1 to get the fifth term: **33**. ### Fibonacci-type sequences A Fibonacci-type sequence makes each new term by adding the two previous terms. The sequence begins 3, 4, 7, 11, 18. Find the next two terms. 1. Add the last two terms: 11 + 18 = 29. 2. The next term is 29. 3. Add the new last two terms: 18 + 29 = 47. 4. The next two terms are **29 and 47**. ## 3. The nth term: using a formula The nth term is a formula that tells you the term in position $n$. The letter $n$ stands for the position number. So $n=1$ means the first term, $n=2$ means the second term, and so on. To use an nth-term formula, substitute the position number into the formula. The nth term of a sequence is $5n + 2$. Find the first two terms, and decide whether 37 is a term. 1. For the first term, use $n=1$: $5(1)+2=7$. 2. For the second term, use $n=2$: $5(2)+2=12$. 3. The first two terms are **7 and 12**. 4. To check whether 37 is a term, solve $5n + 2 = 37$: $$ \begin{aligned} 5n + 2 &= 37 \\ 5n &= 35 \\ n &= 7 \end{aligned} $$ 5. Since 7 is a positive whole-number position, **37 is a term**. In GCSE sequence questions, the first term is normally when $n=1$, not when $n=0$, unless the question says otherwise. ### Formulae with squares If the formula contains $n^2$, square the position number first. The nth term is $n^2 + 5$. Find the first three terms, and decide whether 54 is a term. 1. Use $n=1$: $1^2+5=6$. 2. Use $n=2$: $2^2+5=9$. 3. Use $n=3$: $3^2+5=14$. 4. To check 54, solve $n^2 + 5 = 54$: $$ \begin{aligned} n^2 + 5 &= 54 \\ n^2 &= 49 \\ n &= 7 \end{aligned} $$ 5. Since 7 is a positive whole-number position, **54 is a term**. ## 4. Finding the nth term of an arithmetic sequence An **arithmetic sequence** has the same difference between consecutive terms. Consecutive terms are terms next to each other. The fixed difference is called the **common difference**. For an arithmetic sequence, the nth term has the form $dn + c$. - $d$ is the common difference. - $c$ is the adjustment needed to match the actual sequence. The number in front of $n$ comes from the common difference, not from the first term. Find the nth term of 8, 13, 18, 23, 28. 1. The common difference is 5, because each term increases by 5. 2. Start with $5n$. Its first few values are 5, 10, 15, 20, 25. 3. Compare with the actual sequence: 8, 13, 18, 23, 28. 4. Each actual term is 3 bigger than the matching $5n$ value. 5. So the nth term is **$5n + 3$**. ### Decreasing arithmetic sequences If the sequence goes down, the common difference is negative. Find the nth term of 26, 22, 18, 14, 10. 1. The sequence goes down by 4 each time, so the common difference is negative 4. 2. Start with $-4n$. Its first few values are negative 4, negative 8, negative 12. 3. To get from negative 4 to 26, add 30. 4. So the nth term is **$-4n + 30$**. 5. Check: when $n=1$, $-4n+30=26$. After finding an nth term, test it with $n=1$ and $n=2$. If it does not give the first two terms, fix it before moving on. ## 5. Pattern sequences Pattern questions work like number sequences. The **pattern number** is the position number. Count carefully, then look for what changes each time.