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1. Sequences: explicit and recursive formulas

Warm-up

Three quick questions from earlier lessons.

Discover

A sequence starts $5, 8, 11, 14, \dots$ Which rule gives the next term from the one before?

Add 3
Multiply by 3

Key idea

A sequence is a function whose inputs are the whole numbers $1, 2, 3, \dots$ : $a_n$ is its $n$ th term.

An explicit formula gives

a_n

from

n

directly (for example

a_n = 3n + 2

). A recursive formula gives a first term and a rule from one term to the next (for example

a_1 = 5

a_{n+1} = a_n + 3

Worked example

$a_1 = 2$ and $a_{n+1} = 3a_n - 1$ . Find $a_4$ .

a_2 = 3 \cdot 2 - 1 = 5

Apply the rule to a₁.

a_3 = 3 \cdot 5 - 1 = 14

Then to a₂.

a_4 = 3 \cdot 14 - 1 = 41

Then to a₃.

Your turn, guided

Your turn: $a_n = n^2 - 1$ . Find $a_5$ .

a_5 = 5^2 - 1

Replace n by 5.

a_5 = 24

25 − 1.

Practice ★

$a_n = n^2 + 5$ . Complete the table.
$a_n = n^2 + 9$ . Complete the table.

Watch out!

Practice ★★

$a_1 = 3$ and $a_{n+1} = 2a_n - 7$ . Find $a_3$ .
$a_1 = 6$ and $a_{n+1} = 2a_n - 3$ . Find $a_3$ .
$a_1 = 4$ and $a_{n+1} = 2a_n - 6$ . Find $a_3$ .

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Summary card

Sequences

Explicit: $a_n$ straight from $n$ .
Recursive: a first term and a rule from $a_n$ to $a_{n+1}$ .

Watch out: check your formula with $n = 1$ .

2. Arithmetic sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A taxi charges 3 dollars, plus 2 dollars per mile. Mile by mile, the fare is $5, 7, 9, \dots$ What does the 10th mile cost in total?

23 dollars
25 dollars

Key idea

An arithmetic sequence adds the same number $d$ (the common difference) each time.

a_n = a_1 + (n - 1)d

. From

a_p

a_q

there are

q - p

steps, so

d = \dfrac{a_q - a_p}{q - p}

Worked example

An arithmetic sequence has $a_3 = 11$ and $a_8 = 31$ . Find $d$ and $a_{20}$ .

d = \dfrac{31 - 11}{5} = 4

5 steps from a₃ to a₈.

a_1 = 11 - 2 \cdot 4 = 3

Back 2 steps from a₃.

a_{20} = 3 + 19 \cdot 4 = 79

19 steps from a₁.

Your turn, guided

Your turn: $a_1 = 7$ , $d = 5$ . Find $a_{12}$ .

a_{12} = 7 + 11 \cdot 5

11 steps.

a_{12} = 62

7 + 55.

Practice ★

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .
An arithmetic sequence has $a_1 = 20$ and common difference 5. Find $a_{ 28 }$ .

Watch out!

Practice ★★

In an arithmetic sequence, $a_{ 4 } = 28$ and $a_{ 9 } = 68$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 18$ and $a_{ 7 } = 34$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 11$ and $a_{ 10 } = 25$ . Find the common difference.

What kind of sequence is $3, 6, 12, 24, \dots$ ?
1. Geometric, with ratio 2
2. Arithmetic
3. Neither

Exit ticket

Three questions to check you have got it.

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .

Summary card

Arithmetic sequences

Add $d$ each time: $a_n = a_1 + (n - 1)d$ .
$d = \dfrac{a_q - a_p}{q - p}$ .

Watch out: $n - 1$ steps from $a_1$ to $a_n$ .

3. Geometric sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A rumor spreads: 3 people know it on day 1, and each day the number triples. How many know it on day 4?

Key idea

A geometric sequence multiplies by the same number $r$ (the common ratio) each time.

a_n = a_1 \cdot r^{\,n-1}

, and

r = \dfrac{a_{n+1}}{a_n}

Worked example

Find $a_7$ for $2, 6, 18, \dots$

r = 3

6 ÷ 2.

a_7 = 2 \cdot 3^6

6 multiplications from a₁ to a₇.

a_7 = 1458

3⁶ = 729.

Your turn, guided

Your turn: $a_1 = 5$ , $r = 2$ . Find $a_6$ .

a_6 = 5 \cdot 2^5

5 multiplications.

a_6 = 160

5 · 32.

Practice ★

A geometric sequence has $a_1 = 3$ and ratio 3. Find $a_{ 4 }$ .
A geometric sequence has $a_1 = 1$ and ratio 2. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 2$ and ratio 3. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 4$ and ratio 2. Find $a_{ 5 }$ .

Watch out!

Practice ★★

Find the common ratio of the geometric sequence $8, 32, 128, \dots$
Find the common ratio of the geometric sequence $7, 21, 63, \dots$

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Summary card

Geometric sequences

Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$ .
$r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$ .

4. Sigma notation and arithmetic series

Warm-up

Three quick questions from earlier lessons.

Discover

To add $1 + 2 + \dots + 100$ , pair the first and last terms: $1 + 100$ , $2 + 99$ , … How many pairs, and what is each pair worth?

50 pairs of 101
100 pairs of 101

Key idea

$\displaystyle\sum_{k=p}^{q} a_k$ means $a_p + a_{p+1} + \dots + a_q$ : it has $q - p + 1$ terms.

The sum of the

n

terms of an arithmetic sequence is

S_n = \dfrac{n(a_1 + a_n)}{2}

: the number of terms times the average of the first and last. In particular

1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}

Worked example

Find $\displaystyle\sum_{k=1}^{20} (3k + 2)$ .

First term

5

, last term

62

k = 1 and k = 20.

S = \dfrac{20(5 + 62)}{2}

20 terms.

S = 670

10 · 67.

Your turn, guided

Your turn: $2 + 4 + 6 + \dots + 40$ .

n = 20

40 ÷ 2: 20 terms.

S = 420

20 · 42 / 2.

Practice ★

How many terms does $\displaystyle\sum_{k=3}^{ 17 } (2k + 1)$ have?
How many terms does $\displaystyle\sum_{k=3}^{ 21 } (2k + 1)$ have?
How many terms does $\displaystyle\sum_{k=2}^{ 38 } (2k + 1)$ have?

Watch out!

Practice ★★

Find $\displaystyle\sum_{k=1}^{ 30 } (3k + 7)$ .
Find $\displaystyle\sum_{k=1}^{ 11 } (6k + 8)$ .
Find $\displaystyle\sum_{k=1}^{ 21 } (6k + 6)$ .

Find $1 + 2 + 3 + \dots + 94$ .
Find $1 + 2 + 3 + \dots + 80$ .

Exit ticket

Three questions to check you have got it.

Find $\displaystyle\sum_{k=1}^{ 30 } (3k + 7)$ .
Find $\displaystyle\sum_{k=1}^{ 11 } (6k + 8)$ .
Find $\displaystyle\sum_{k=1}^{ 21 } (6k + 6)$ .

Summary card

Arithmetic series

$\displaystyle\sum_{k=p}^{q}$ has $q - p + 1$ terms.
$S_n = \dfrac{n(a_1 + a_n)}{2}$ .
$1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$ .

Watch out: count the terms with $+1$ .

5. Finite geometric series

Warm-up

Three quick questions from earlier lessons.

Discover

$S = 1 + 2 + 4 + 8 + 16$ . What is $2S - S$ ?

$32 - 1 = 31$
$16$

Key idea

Multiplying by $r$ and subtracting gives the sum of a geometric sequence.

The sum of the first

n

terms of a geometric sequence (

r \ne 1

) is

S_n = a_1\,\dfrac{r^n - 1}{r - 1}

Worked example

Find $3 + 6 + 12 + \dots$ (8 terms).

S_8 = 3 \cdot \dfrac{2^8 - 1}{2 - 1}

a₁ = 3, r = 2, n = 8.

S_8 = 3 \cdot 255 = 765

2⁸ = 256.

Your turn, guided

Your turn: $1 + 3 + 9 + \dots$ (5 terms).

S_5 = \dfrac{3^5 - 1}{3 - 1}

a₁ = 1, r = 3.

S_5 = 121

(243 − 1)/2.

Practice ★

Find the sum of the first 6 terms of the geometric sequence with $a_1 = 8$ and ratio 3.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 7$ and ratio 2.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 4$ and ratio 2.

Watch out!

Practice ★★

Find $\displaystyle\sum_{k=0}^{ 6 } 2^k$ .
Find $\displaystyle\sum_{k=0}^{ 10 } 2^k$ .

You put 764 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 12th payment? Round to the nearest cent.
You put 344 dollars in a savings account at the end of each year, at 5 % interest a year. How much is in the account right after the 11th payment? Round to the nearest cent.

Exit ticket

Three questions to check you have got it.

Find the sum of the first 6 terms of the geometric sequence with $a_1 = 8$ and ratio 3.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 7$ and ratio 2.
Find the sum of the first 5 terms of the geometric sequence with $a_1 = 4$ and ratio 2.

Summary card

Finite geometric series

$S_n = a_1\,\dfrac{r^n - 1}{r - 1}$ .
$n$ is the number of terms; the power is $r^n$ .

Watch out: the last term is $a_1 r^{\,n-1}$ , but the sum uses $r^n$ .

6. Infinite geometric series

Warm-up

Three quick questions from earlier lessons.

Discover

You walk half way to a wall, then half the remaining distance, and so on: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dots$ What does this add up to?

1
It grows without bound

Key idea

When $|r| < 1$ , the terms shrink so fast that the infinite sum has a value.

|r| < 1

a + ar + ar^2 + \dots = \dfrac{a}{1 - r}

. If

|r| \ge 1

, the series has no sum.

Worked example

Write $0.\overline{36} = 0.363636\dots$ as a fraction.

0.\overline{36} = \dfrac{36}{100} + \dfrac{36}{100^2} + \dots

Blocks of two digits.

S = \dfrac{36/100}{1 - 1/100} = \dfrac{36}{99}

a = 36/100, r = 1/100.

0.\overline{36} = \dfrac{4}{11}

Divide by 9.

Your turn, guided

Your turn: $6 + 2 + \dfrac{2}{3} + \dots$

r = \dfrac{1}{3}

, and

|r| < 1

2 ÷ 6.

S = 9

6 / (1 − 1/3) = 6 / (2/3).

Practice ★

Find the sum $7 + \dfrac{ 7 }{ 5 } + \dfrac{ 7 }{ 25 } + \dots$
Find the sum $2 + \dfrac{ 2 }{ 5 } + \dfrac{ 2 }{ 25 } + \dots$
Find the sum $9 + \dfrac{ 9 }{ 5 } + \dfrac{ 9 }{ 25 } + \dots$

Watch out!

Practice ★★

Does the series $1 + \dfrac{ 6 }{ 2 } + \left(\dfrac{ 6 }{ 2 }\right)^2 + \dots$ have a sum?
1. No: the ratio is greater than 1, so the terms keep growing
2. Yes: $\dfrac{1}{1 - r}$ with $r = \dfrac{ 6 }{ 2 }$
3. Yes: every infinite geometric series has a sum

Write $0.\overline{ 4 } = 0.444\dots$ as a fraction, using $\dfrac{ 4 }{10} + \dfrac{ 4 }{100} + \dots$
Write $0.\overline{ 6 } = 0.666\dots$ as a fraction, using $\dfrac{ 6 }{10} + \dfrac{ 6 }{100} + \dots$
Write $0.\overline{ 1 } = 0.111\dots$ as a fraction, using $\dfrac{ 1 }{10} + \dfrac{ 1 }{100} + \dots$

Exit ticket

Three questions to check you have got it.

Find the sum $7 + \dfrac{ 7 }{ 5 } + \dfrac{ 7 }{ 25 } + \dots$
Find the sum $2 + \dfrac{ 2 }{ 5 } + \dfrac{ 2 }{ 25 } + \dots$
Find the sum $9 + \dfrac{ 9 }{ 5 } + \dfrac{ 9 }{ 25 } + \dots$

Summary card

Infinite geometric series

$|r| < 1$ : $S = \dfrac{a}{1 - r}$ .
$|r| \ge 1$ : no sum.
Repeating decimals are such series.

Watch out: check $|r| < 1$ before using the formula.

7. The binomial theorem

Warm-up

Three quick questions from earlier lessons.

Discover

$(1 + x)^2 = 1 + 2x + x^2$ and $(1 + x)^3 = 1 + 3x + 3x^2 + x^3$ . The coefficients $1, 3, 3, 1$ are a row of Pascal's triangle. What is the next row?

$1, 4, 6, 4, 1$
$1, 4, 4, 4, 1$

Key idea

The binomial theorem expands any power of a sum.

(a + b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k}

, with

\dbinom{n}{k} = \dfrac{n!}{k!\,(n - k)!}

. For example

\dbinom{n}{1} = n

and

\dbinom{n}{2} = \dfrac{n(n - 1)}{2}

Worked example

Find the coefficient of $x^2$ in $(x + 3)^5$ .

Term:

\dbinom{5}{3} x^2 \cdot 3^3

x² means b = 3 appears 3 times: k = 3 with b = 3.

10 \cdot 27 = 270

C(5, 3) = 10.

Your turn, guided

Your turn: the coefficient of $x^2$ in $(1 + x)^6$ .

\dbinom{6}{2} = \dfrac{6 \cdot 5}{2}

C(6, 2).

= 15

30 ÷ 2.

Practice ★

Find the coefficient of $x^2$ in $(1 + x)^{ 10 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 7 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 5 }$ .

Watch out!

Practice ★★

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

In $(2x + 2)^2 = 4x^2 + kx + 4$ , what is $k$ ?
In $(7x + 2)^2 = 49x^2 + kx + 4$ , what is $k$ ?

Exit ticket

Three questions to check you have got it.

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

Summary card

The binomial theorem

$(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$ .
Coefficients: a row of Pascal's triangle.
Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$ .

Answers: Sequences and series

1. Sequences: explicit and recursive formulas

$6, 9, 14, 21$
$10, 13, 18, 25$

$-9$
$15$
$-2$

$5 n - 4$
$7 n - 4$
$6 n + 2$

$5 n - 4$
$7 n - 4$
$6 n + 2$

2. Arithmetic sequences

$98$
$86$
$133$
$155$

Geometric, with ratio 2

$98$
$86$
$133$

3. Geometric sequences

$81$
$128$
$4374$
$64$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$

4. Sigma notation and arithmetic series

$15$
$19$
$37$

$1605$
$484$
$1512$

$4465$
$3240$

$1605$
$484$
$1512$

5. Finite geometric series

$2912$
$217$
$124$

$127$
$2047$

$\frac{622627054674848431}{51200000000000}$
$\frac{6255532523315503}{1280000000000}$

$2912$
$217$
$124$

6. Infinite geometric series

$\frac{35}{4}$
$\frac{5}{2}$
$\frac{45}{4}$

No: the ratio is greater than 1, so the terms keep growing

$\frac{4}{9}$
$\frac{2}{3}$
$\frac{1}{9}$

$\frac{35}{4}$
$\frac{5}{2}$
$\frac{45}{4}$

7. The binomial theorem

$45$
$21$
$10$

$256$
$405$
$500$

$8$
$28$

$256$
$405$
$500$