A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 3: Sequences and series · Lesson 1

Arithmetic sequences and series

113 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-arithmetic-sequences-and-series/

Date

Do Now

Quick questions to start. They use what you need today.

A1
Needed today
Solve $n^{2} + 3 n - 130 = 0$ , where $n$ is a positive integer.
[2 marks]
Answer
A2
Needed today
Find an expression for the $n$ th term of $7, 11, 15, 19, \dots$
[2 marks]
Answer
A3
Common slip
An arithmetic sequence has first term $5$ and common difference $3$ . Find the 20th term.
[2 marks]
Answer
A4
Needed today
Find the smallest positive integer $n$ for which $n^{2} - 9 n - 52 > 0$ .
[3 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

The arithmetic sequence $2, 9, 16, 23, \dots$ has $n$ th term $u_{n}$ . Find $u_{30}$ , and find which term is equal to $359$ .

Hint Use

u_{n} = a + (n - 1) d

. For the position, set

u_{n} = 359

and solve for

n

B1
The arithmetic sequence $85, 81, 77, \dots$ has $n$ th term $u_{n}$ . (a) Find $u_{25}$ . (b) Find which term is equal to $- 51$ .
[4 marks]
Answer

Example 2

In an arithmetic series $u_{5} = 23$ and $u_{12} = 51$ . Find the first term, the common difference and $S_{20}$ .

Hint Write two equations in

a

and

d

and subtract them.

B2
In an arithmetic series $u_{4} = 17$ and $u_{15} = - 27$ . Find the first term, the common difference and $S_{25}$ .
[5 marks]
Answer

Example 3

Prove that the sum of the first $n$ terms of an arithmetic series with first term $a$ and common difference $d$ is $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ .

Hint Write the series forwards, then backwards underneath, and add the two lines term by term.

B3
Prove that $1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}$ .
[4 marks]

Example 4

For the series $3 + 7 + 11 + \dots$ , find the least number of terms for which the sum is greater than $500$ .

Hint Write

S_{n}

as a quadratic in

n

, solve

S_{n} = 500

, then round to the next whole number. Check the two values either side.

B4
For the series $50 + 46 + 42 + \dots$ , find the least number of terms for which the sum is negative.
[5 marks]
Answer

Example 5

Find the sum of all the multiples of $7$ between $100$ and $500$ .

Hint Find the first and last multiples, then the number of terms, before you use a sum formula.

B5
Find the sum of all the multiples of $9$ between $200$ and $800$ .
[5 marks]
Answer

Practice

A-Level

C1
A student finds the sum of the first $15$ terms of $8 + 5 + 2 + \dots$ and writes $S_{15} = \frac{15}{2} (2 \times 8 + 15 \times (- 3)) = - 217.5$ . Explain the mistake and find the correct value of $S_{15}$ .
[4 marks]
Answer $S_{15} = \dots\dots$

C2
Find the sum of the integers from $1$ to $100$ that are not multiples of $4$ .
[5 marks]
Answer
C3
The sum of the first $n$ terms of a series is $S_{n} = 3 n^{2} + 2 n$ . Find an expression for $u_{n}$ , show that the series is arithmetic, and find $u_{10}$ .
[5 marks]
Answer
C4
The sum of the first $n$ terms of $4 + 7 + 10 + \dots$ is $1079$ . Find $n$ .
[4 marks]
Answer
C5
Find the sum of the series $7 + 11 + 15 + \dots + 403$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The first three terms of an arithmetic sequence are $(k + 4)$ , $(3 k - 1)$ and $(4 k + 5)$ , where $k$ is a constant.
(a)
Find the value of $k$ .
[2 marks]
Answer
(b)
Find the 20th term of the sequence.
[2 marks]
Answer
(c)
Find the least value of $n$ for which the sum of the first $n$ terms is greater than $10000$ .
[5 marks]
Answer
Total for D1: 9 marks

D2
Raj wants the greatest $n$ for which the sum of the first $n$ terms of $120 + 113 + 106 + \dots$ is positive. He writes " $S_{n} = \frac{n}{2} (240 - 7 n)$ , so $S_{n} > 0$ when $n < \frac{240}{7} = 34.3$ . The greatest value is $n = 34$ ."
Raj is wrong.
(a)
Find Raj's mistake and give the correct greatest $n$ .
[4 marks]
(b)
Find the greatest value of $S_{n}$ , and the value of $n$ at which it occurs.
[4 marks]
Answer
Total for D2: 8 marks
D3
The sum of the first $10$ terms of an arithmetic series is $175$ . The sum of the next $10$ terms is $575$ . Find the first term and the common difference.
[7 marks]
Answer
D4
The first three terms of a sequence are $lo g_{2} 3$ , $lo g_{2} 12$ and $lo g_{2} 48$ .
(a)
Show that the sequence is arithmetic and state the common difference.
[3 marks]
(b)
Show that $S_{10} = 10 lo g_{2} 3 + 90$ .
[3 marks]
(c)
Find the least value of $n$ for which $S_{n} > 200$ .
[4 marks]
Answer
Total for D4: 10 marks
D5
Sequence $A$ is $5, 12, 19, \dots$ and sequence $B$ is $203, 199, 195, \dots$ . Both are arithmetic.
(a)
Show that the 19th terms of $A$ and $B$ are equal.
[3 marks]
(b)
Find the least value of $n$ for which the sum of the first $n$ terms of $A$ is greater than the sum of the first $n$ terms of $B$ .
[6 marks]
Answer
Total for D5: 9 marks
D6
Show that the sum of all the three-digit numbers that leave a remainder of $2$ when divided by $5$ is $98910$ .
[7 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
An arithmetic series has first term $1$ and common difference $d$ , where $d$ is a positive integer. For some $n \geq 2$ , the sum of the first $n$ terms is $1000$ . Find every possible value of $d$ .
[9 marks]
Hint 1 · What to try
Write $S_{n} = 1000$ and clear the fraction: $n (2 + (n - 1) d) = 2000$ .
Hint 2 · The first line
So $n$ must be a factor of $2000$ , and $\frac{2000}{n} - 2$ must divide exactly by $n - 1$ .
Hint 3 · The full method
Since $d \geq 1$ , $n (n + 1) \leq 2000$ , so $n \leq 44$ . Check each factor of $2000$ up to $44$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the $n$ th term of an arithmetic sequence and which term has a given value.
I can find the first term and common difference from two facts.
I can prove the formula for the sum of an arithmetic series.
I can find the least or greatest $n$ for a condition on $S_{n}$ .

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers