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This page sets out the method for sequences (nth term) in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Sequences (nth term) topic page.
When you are asked for the nth term of a sequence that rises or falls steadily.
When you are asked whether a number is a term, set the nth term equal to it.
In a linear sequence the difference between terms is constant.
3, 7, 11, 15 goes up by 4 each time.
The nth term is (difference) × n, then add or subtract a constant.
Here it is 4n − 1, because 4 × 1 − 1 = 3.
Put n = 1, 2 and 3 back into your rule.
You should get the start of the sequence, not the difference.
The 10th term is the rule with n = 10.
For 4n − 1 the 10th term is 4 × 10 − 1 = 39.
A sequence that falls has a negative difference.
20, 17, 14 has difference −3, so the nth term starts with −3n.
If the differences themselves change, the rule is not of the form an + b.
This topic is the linear case.
For 5, 8, 11, 14 the gap is +3. The nth term starts as 3n, not n+3. "Goes up by 3" describes a multiplication of the position, because each new term adds another 3. The sequence n+3 is 4, 5, 6, 7, which is a different list. Write the difference down before you write any algebra.
Compare 3n with the real sequence. 3× 1=3, but the first term is 5, which is 2 more. So the rule is 3n+2. That +2 is the zero term: one step before the first term would be 5-3=2. In general, constant = first term - difference. Then test n=1, n=2 and n=3. All three should match. One match can be luck.
Method: 1. For the terms 5, 8, 11, 14, the common gap is +3. 2. Begin the nth term as 3n, not as n+3. 3. A rise of 3 at each step means you multiply the position by 3.
A sequence that falls has a negative difference. 40, 33, 26, 19 falls by 7, so the coefficient is -7. The zero term is 40-(-7)=47. The nth term is 47-7n. Writing 40-7n uses the first term as the constant and is already wrong at n=1: 40-7× 1=33, which is the second term, not the first. Writing 7n+33 gives 40 at n=1, but it rises: at n=2 it gives 47, not 33.
You can also write -7n+47. Both forms are the same expression. Check n=1: 47-7=40. Check n=4: 47-28=19. A negative coefficient is not a sign that the method has failed. It is what "decreasing" looks like in algebra.
Method: 1. Use a negative coefficient when the terms get smaller. 2. The sequence 40, 33, 26, 19 falls by 7, so the coefficient is -7. 3. The zero term is 40-(-7), which equals 47.
The 50th term of 3n+2 is 3× 50+2=152. Substitute the position, not the term value. Asking "is 200 a term?" is the opposite substitution: set the expression equal to 200 and solve for n. 3n+2=200, so 3n=198 and n=66. Because 66 is a positive whole number, 200 is the 66th term. If n had come out as 66.5 or as a negative, 200 would not be in the sequence.
The sentence that earns the mark is "n is an integer" or "n is not an integer". A long list of terms is slower and easier to miscount. Pattern questions about matches or tiles hide a sequence in a picture. Count the first three pictures, find the difference, and build the nth term in the usual way. The picture does not change the algebra.
Method: 1. Substitute the position 50 into 3n+2 to get 3× 50+2=152. 2. Substitute the position into the expression. Do not put the term value in as n. 3. To test whether 200 is a term, set the expression equal to 200 and solve for n.
Exam questions often hide a sequence inside a picture or a price list. A pattern of tiles, matchsticks or chairs gives you the number of items in pattern 1, 2, 3 and so on. Treat those counts as the terms of a sequence. The pattern number is n and the number of items is the term. Write the first few terms down in a row before you do anything else.
A fixed amount plus a repeated amount is a very common context. A bike hire shop that charges a set fee plus a price per hour makes a linear sequence. The price per hour is the difference, so it is the number in front of n. The fixed fee is what is left over when you take the zero term. Always say what n stands for before you write the rule.
Method: 1. Write the counts for pattern 1, 2, 3 as a list of terms. 2. Find the difference between neighbouring terms. This is the number multiplying n. 3. Work back to the zero term by subtracting the difference from the first term. 4. Write the rule as an + b and test it on the first two terms.
Tip: In a costs question, the repeated charge is the difference and the one-off charge is the number added on.
To decide whether a number belongs to a sequence, set the nth term equal to that number and solve for n. If n comes out as a whole number, the number is a term and n tells you its position. If n is a fraction or a decimal, the number is not in the sequence. Position numbers are always whole numbers, so you cannot have the 12.25th term.
The question usually says Explain or Show that, and the marks go for the working, not just the word yes or no. Write the equation, solve it, and finish with a sentence such as "n is not a whole number, so it is not a term". A second way to explain is to look at what every term has in common, for example "every term is one more than a multiple of 4". Use whichever you can justify clearly.
Method: 1. Write the nth term equal to the number you are testing. 2. Solve the equation to find n. 3. Look at whether n is a positive whole number. 4. Write a conclusion that mentions n and the words "is a term" or "is not a term".
Tip: A bare "no" earns nothing. The mark is for the equation and the reason.
Harder questions give two sequences and ask where they meet. One kind asks for a number that appears in both lists. List enough terms of each sequence until the same value appears twice, then keep going to find the next one. Common values repeat at a steady gap, which is a quick check on your list.
Another kind gives two nth terms and asks for the value of n where the two terms are equal. Here you write the two expressions with an equals sign between them and solve. For 5n - 3 = 2n + 9 collect the n terms on one side, divide, and then substitute back to get the actual term. Say clearly whether the question wants n or the term, because they are usually different numbers.
Method: 1. Decide whether the question wants a shared term or the shared position n. 2. For shared terms, list the first several terms of both sequences side by side. 3. For the same position, set the two nth terms equal and solve the equation. 4. Substitute back into either rule to find the actual term, then check it in the other rule.
Some questions ask for the first term that is bigger than a number, or how many terms lie between two numbers. Treat these as inequalities. Write the rule as greater than or less than the target and solve to get a boundary for n. Then choose the next whole number that keeps the inequality true.
The common trap is rounding in the wrong direction. If 7n - 4 > 100 gives n > 14.86, then n = 14 is too small and n = 15 is the first position that works. Always check by substituting your whole number and its neighbour. To count terms from position 15 to position 29 inclusive, work out 29 - 15 + 1, not just 29 - 15.
Method: 1. Write the nth term with the correct inequality sign and solve for n. 2. Take the next whole number in the direction that keeps the inequality true. 3. Substitute that whole number and the one before it to check. 4. To count terms, use last position minus first position plus one.
Tip: Count positions with "last minus first, plus one". Terms 15 to 29 is 15 terms.
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