Sequences and nth term – Year 9 maths worksheet with answers to download as PDF

A term-to-term rule tells you how to get from one term to the next. Find the rule and write the next two terms of each sequence.

1. \(3,7,11,15,\ldots\)
2. \(100,90,80,70,\ldots\)
3. \(2,6,18,54,\ldots\)
4. \(1,4,9,16,\ldots\)

The nth term formula gives the term in position \(n\). Substitute \(n=1,2,3\ldots\) to find terms.

1. \(3n+2\): write the first four terms and the 10th term.
2. \(n^2+1\): write the first four terms and the 5th term.
3. \(50-4n\): write the first three terms and the 12th term.
4. \(2^n\): write the first five terms.

For an arithmetic sequence the nth term is \(dn+(a-d)\), where \(d\) is the common difference and \(a\) is the first term. Find the nth term of each sequence.

1. \(5,8,11,14,\ldots\)
2. \(4,10,16,22,\ldots\)
3. \(20,17,14,11,\ldots\)
4. \(7,12,17,22,\ldots\)

To check whether a number is in a sequence, set the nth term equal to the number and solve. It is in the sequence only if \(n\) is a positive whole number.

1. Is 100 a term of \(3n+1\)?
2. Is 150 a term of \(3n+1\)?
3. Is 63 a term of \(5n-2\)?
4. Is 80 a term of \(5n-2\)?

In a geometric sequence you multiply by the same number, the common ratio \(r\), each time.

1. \(3,6,12,24,\ldots\) Find \(r\) and the next two terms.
2. \(81,27,9,3,\ldots\) Find \(r\) and the next two terms.
3. The sequence \(5,10,20,\ldots\) has nth term \(5\times2^{n-1}\). Find the 8th term.
4. Is \(3,6,9,12\) geometric? Explain.

Some sequences have famous names. The square numbers are \(1,4,9,16,\ldots\), the cube numbers are \(1,8,27,64,\ldots\) and the triangular numbers are \(1,3,6,10,\ldots\) In a Fibonacci-type sequence each term is the sum of the two terms before it.

1. Write the next three triangular numbers.
2. Find the 10th square number and the 6th cube number.
3. A Fibonacci-type sequence starts \(2,3,\ldots\) Write the next three terms.
4. Powers of 2 start \(2,4,8,16,\ldots\) Find the 10th term.

Tom makes a row of squares using matchsticks. One square uses 4 matchsticks, two squares use 7, and three squares use 10.

1. Write down the number of matchsticks for 4 squares and 5 squares.
2. Find the nth term for the number of matchsticks for \(n\) squares.
3. How many matchsticks are needed for 20 squares?
4. Tom has 151 matchsticks. How many squares can he make?

Ruth has £15 in her money box and adds £8 at the end of every week. Let the amount after \(n\) weeks be \(A_n\).

1. Find the nth term for the amount.
2. How much will she have after 12 weeks?
3. After how many weeks will she first have at least £200?
4. Her brother has £60 and saves £5 a week. When does Ruth first have more than her brother?

A school theatre has 12 seats in the first row. Each row behind has 2 more seats than the row in front.

1. How many seats are in rows 2, 3 and 4?
2. Find the nth term for the number of seats in row \(n\).
3. How many seats are in row 15?
4. There are 8 rows altogether. Find the total number of seats.

A ball is dropped from 160 cm. Each time it bounces it reaches half the height of the previous bounce.

1. What type of sequence is this? State the common ratio.
2. Find the height of the 6th bounce.
3. Write an expression for the height of the \(n\)th bounce.
4. After which bounce is the height first below 1 cm?

Mrs Evans is building a square pond from 1 slab, then 2 by 2 slabs, then 3 by 3 slabs. She surrounds each pond with a border of black tiles one tile wide, as in the table.

1. Find the nth term for the number of black tiles around an \(n\) by \(n\) pond.
2. How many black tiles are needed for a 10 by 10 pond?
3. A pond needs 84 black tiles. What size is it?
4. Explain the formula by looking at the corners and the sides of the border.

A pair of rabbits is bred. In the Fibonacci model, the number of pairs each month is the sum of the numbers in the previous two months. The first two months have 1 pair each.

1. Find the number of pairs in months 7, 8 and 9.
2. Find the number in month 10.
3. Does this sequence have a constant difference or a constant ratio? Explain.
4. The ratios of consecutive terms get close to 1.6. Check this for months 9 and 10.

A supermarket stacks tins in a triangle: 1 tin in the top row, 2 in the second, 3 in the third, and so on. The numbers of tins in the whole stack are the triangular numbers \(1,3,6,10,\ldots\)

1. Write the number of tins in a stack with 4 rows and with 8 rows.
2. The formula for the number in a stack with \(n\) rows is \(\dfrac{n(n+1)}2\). Check it for \(n=4\).
3. How many tins are in a 10-row stack?
4. A stack has 45 tins. How many rows does it have?

Hamza starts a running plan. In week 1 he runs 2 km. Every week after that he adds 0.5 km to the length of his weekly long run.

1. Find the nth term for the distance in week \(n\).
2. How far does he run in week 10?
3. In which week will he first run 10 km or more?
4. After the plan ends he runs 5 km a week. Is it likely that the formula keeps working for ever? Explain.

A teacher gives four sequences. Complete the table.

1. Complete the table.
2. Which sequence is not arithmetic? How can you tell?
3. Which sequence has a negative 50th term?
4. For which value of \(n\) does the third sequence first become negative?

Sequence A has nth term \(3n+2\) and sequence B has nth term \(5n-3\).

1. Which number appears in both sequences in the table?
2. Show that 32 is a term of both sequences and find \(n\) for each.
3. Show that there is no value of \(n\) at which A and B have the same term in the same position.
4. Which sequence grows faster, and why?

Look at the sequence \(3,8,15,24,35,\ldots\)

1. What is special about the second differences?
2. The nth term has the form \(n^2+bn\). Use the first term to find \(b\).
3. Check your formula with \(n=4\).
4. Find the 10th term.
5. Is 99 a term of the sequence?

An arithmetic sequence has 4th term 17 and 9th term 42.

1. Find the common difference.
2. Find the first term and the nth term.
3. Find the 20th term.
4. Is 497 a term of the sequence? Is 500?

A chain letter is started by one person on day 1. Each person who receives it sends it to 3 new people the next day. The numbers who receive it each day are \(1,3,9,27,\ldots\)

1. Find the common ratio and the nth term.
2. How many receive the letter on day 8?
3. The population of a town is 100 000. On which day does the number who receive the letter first exceed this?
4. Explain why the chain cannot really continue for ever.

The triangular numbers are \(T_n=\dfrac{n(n+1)}2\): \(1,3,6,10,15,\ldots\) and the square numbers are \(n^2\).

1. Write down \(T_1\) to \(T_5\).
2. Show that \(T_4+T_5=25\).
3. Show by algebra that \(T_n+T_{n+1}=(n+1)^2\).
4. Use the result to find \(T_{19}+T_{20}\) and check by using the formula.
5. Find the first triangular number greater than 200.