Predicting What Comes Next: Sequences and Progressions

Contents hide

Chapter mind map: how it all connects

1 · Sequences and rulesAn ordered list of terms; explicit rules use position

n

directly, recursive rules use the previous term

2 · Arithmetic ProgressionsConstant common difference

d

; terms grow in a straight line

Sequences and Progressions

3 · Sum of first n naturalsGauss’s pairing trick gives

S_n = \frac{n(n+1)}{2}

4 · Geometric ProgressionsConstant common ratio

r

; terms curve away, growing or shrinking fast

What you will learn in this chapter

How to describe a sequence with an explicit rule or a recursive rule, and when each is useful
What makes a sequence an Arithmetic Progression (AP), and how to find any term using $t_n = a+(n-1)d$
How to find the sum of the first $n$ natural numbers using Gauss’s pairing method
What makes a sequence a Geometric Progression (GP), and how to find any term using $t_n = ar^{n-1}$

sequenceexplicit rulerecursive rulearithmetic progressioncommon differencegeometric progressioncommon ratio

1Sequences: Terms and Rules

Learn by heartDefinition 1

A sequence is an ordered list of numbers, where each number is called a term. A sequence can be finite (a fixed number of terms) or infinite (continuing forever, shown by “ $\ldots$ “). We write $t_1$ for the first term, $t_2$ for the second term, and in general $t_n$ for the $n$ th term.

Sequence	Rule in words
1, 2, 3, 4, 5, …	Natural numbers: each term is one more than the previous
1, 3, 5, 7, 9, …	Odd numbers: difference of 2 between consecutive terms
1, 3, 6, 10, 15, …	Triangular numbers: each term adds the next natural number
1, 4, 9, 16, 25, …	Square numbers: each term adds the next odd number

Learn by heartDefinition 2

An explicit rule gives the value of $t_n$ directly from its position number $n$ , e.g. $t_n = 2n-1$ . A recursive rule gives $t_n$ in terms of the previous term(s), e.g. $t_1=1,\ t_n = t_{n-1}+3$ for $n \ge 2$ . An explicit rule finds any term instantly; a recursive rule needs every earlier term to be known first.

Given

A sequence with explicit rule $t_n = 3n-2$

Find

The 10th term, and check whether 76 is a term

Solution

$t_{10} = 3(10)-2 = \;$ $28$

Solving $3n-2=76$ gives $3n=78$ , so $n=26$ : 76 is the 26th term

Given

A sequence with recursive rule $u_1=2,\ u_n = 3u_{n-1}-1$ for $n \ge 2$

Find

The first four terms

Solution

$u_2 = 3(2)-1=5$ , $\;u_3 = 3(5)-1=14$ , $\;u_4 = 3(14)-1=41$

First four terms: $2, 5, 14, 41$

Did you know?

The sequence $1, 2, 3, 5, 8, 13, 21, \ldots$ (each term the sum of the previous two) is the Virahānka-Fibonacci sequence. It was first written down by the Indian scholar Virahānka in the 7th century CE, while studying the rhythms of Prakrit poetry, centuries before the Italian mathematician Fibonacci described it around 1200 CE.

2Arithmetic Progressions (AP)

Learn by heartDefinition 3

An Arithmetic Progression (AP) is a sequence in which every term after the first is obtained by adding a fixed number $d$ , the common difference, to the previous term. Its general form is $a,\ a+d,\ a+2d,\ a+3d,\ \ldots$ , where $a$ is the first term.

Formula

n

th term of an AP:

\quad t_n = a + (n-1)\,d

Recursive form:

t_1 = a,\ \ t_n = t_{n-1} + d

for

n \ge 2

Given

An AP: 7, 11, 15, 19, …

Find

The 15th term

Solution

$a=7$ , $d=4$ . $\;t_{15} = 7 + (15-1)(4) = 7+56 = \;$ $63$

Given

A taxi charges a fixed booking fee of ₹200 plus ₹40 per km travelled

Find

The sequence of total fares, and the fare for 10 km

Solution

After 1, 2, 3 km the fares are ₹240, ₹280, ₹320, … an AP with $a=240$ , $d=40$ . Writing distance travelled as $n$ km, $\;t_n = 200+40n$ .

For $n=10$ : $t_{10} = 200+400 = \;$ ₹600

Plotting (stage, value) pairs of an AP always gives points that lie on a straight line, while a GP’s points curve sharply away, as the chart below shows.

Line chart comparing an arithmetic progression 1,5,9,13,17 which lies on a straight line, with a geometric progression 3,6,12,24,48 which curves upward — **Figure 1**: An AP’s points always lie on a straight line; a GP’s points curve away as the terms grow multiplicatively.

Exam Tip

To test if a sequence is an AP, check that $t_2-t_1 = t_3-t_2 = t_4-t_3$ , and so on. If the difference is not constant, it is not an AP.

3Sum of the First $n$ Natural Numbers

Let $S = 1+2+3+\cdots+n$ . Writing the same sum in reverse, $S = n+(n-1)+\cdots+1$ . Adding the two, every pair of matching terms sums to $n+1$ , and there are $n$ such pairs:

Formula

2S = n(n+1) \quad\Rightarrow\quad S_n = \dfrac{n(n+1)}{2}

Gauss’s pairing method: add the sequence to its own reverse

Did you know?

The earliest known written record of this formula appears in Āryabhaṭa’s Āryabhaṭīya (Chapter 2, Verse 19), composed around 499 CE, describing the sum as the average of the first and last term, multiplied by the number of terms.

Given

The sum of the first 40 natural numbers

Find

$S_{40}$ , and the sum of the numbers from 41 to 60

Solution

$S_{40} = \dfrac{40 \times 41}{2} = \;$ $820$

$41+42+\cdots+60 = S_{60}-S_{40} = \dfrac{60\times 61}{2} – 820 = 1830-820 = \;$ $1010$

Note

This same formula gives the $n$ th triangular number: $t_n = \dfrac{n(n+1)}{2}$ , since each triangular number is the sum of the natural numbers up to that point.

4Geometric Progressions (GP)

Learn by heartDefinition 4

A Geometric Progression (GP) is a sequence in which every term after the first is obtained by multiplying the previous term by a fixed number $r$ , the common ratio. Its general form is $a,\ ar,\ ar^2,\ ar^3,\ \ldots$ , where $a$ is the first term.

Formula

n

th term of a GP:

\quad t_n = a\,r^{\,n-1}

Recursive form:

t_1 = a,\ \ t_n = r \cdot t_{n-1}

for

n \ge 2

. Test:

\dfrac{t_2}{t_1} = \dfrac{t_3}{t_2} = \cdots

must be constant

Given

A GP where the 3rd term is 12 and the 6th term is 96

Find

The common ratio and the first term

Solution

$t_3 = ar^2 = 12$ and $t_6 = ar^5 = 96$ . Dividing, $r^3 = \dfrac{96}{12} = 8$ , so $r = \;$ $2$

Then $a = \dfrac{12}{r^2} = \dfrac{12}{4} = \;$ $3$

Given

A ball dropped from 50 m bounces back to 40% of its previous height each time

Find

The height reached after the 3rd bounce

Solution

Heights form a GP with $a = 50 \times 0.4 = 20$ and $r = 0.4$ : $20,\ 8,\ 3.2,\ \ldots$

$t_3 = 20 \times (0.4)^2 = \;$ $3.2$ m

Did you know?

The Sierpiński triangle is built by repeatedly removing the central triangle from each remaining piece. The number of black triangles at stage $n$ is $3^n$ , while the shaded area shrinks as $\left(\frac34\right)^n$ of the original: both are GPs, one growing without bound, the other shrinking towards zero.

An explicit rule lets you jump straight to any term; a recursive rule only lets you walk there one step at a time.

the single idea to carry out of this chapter

Check yourself before the exam

Can I tell the difference between an explicit rule and a recursive rule, and use each one?
Can I find any term of an AP using $t_n = a+(n-1)d$ , and identify $a$ and $d$ ?
Can I find the sum of the first $n$ natural numbers, and use it to sum a range of numbers?
Can I find any term of a GP using $t_n = ar^{n-1}$ , and identify $a$ and $r$ ?

Quick Revision: read this the night before the exam

Explicit rule: $t_n$ from $n$ directly. Recursive rule: $t_n$ from $t_{n-1}$
AP: constant common difference $d$ ; $t_n = a+(n-1)d$ ; points plot on a straight line
$S_n = 1+2+\cdots+n = \dfrac{n(n+1)}{2}$ (Gauss’s pairing method)
GP: constant common ratio $r$ ; $t_n = ar^{n-1}$ ; points curve away, not a straight line
Virahānka-Fibonacci sequence: $t_n = t_{n-1}+t_{n-2}$ , first recorded 7th century CE

Practice Questions

1 markA sequence has explicit rule $t_n = 4n-1$ . Find the first term and the 5th term.
2 marksFor the sequence $t_n = 3n-2$ , which term of the sequence is 76?
2 marksA sequence is given by the recursive rule $u_1=2$ , $u_n = 3u_{n-1}-1$ for $n \ge 2$ . Find the first four terms.
2 marksFind the 15th term of the AP 7, 11, 15, 19, …
2 marksWhich term of the AP 100, 93, 86, … is 2?
3 marksFind the sum of the first 40 natural numbers. Hence find the sum of the numbers from 41 to 60.
2 marksHow many 2-digit numbers are divisible by 4? Find their sum.
2 marksShow that 2, 6, 18, 54, … is a GP, and find its 6th term.
3 marksIn a GP, the 3rd term is 12 and the 6th term is 96. Find the common ratio and the first term.
3 marksA ball is dropped from a height of 50 m and bounces back to 40% of its previous height each time. Find the height it reaches after the 3rd bounce, and write the explicit formula for the height after the $n$ th bounce.

Multiple Choice

1 markThe common difference of the AP 5, 2, −1, −4, … is:
(a) 3(b) −3(c) 2(d) −2
1 markThe $n$ th term of a GP with first term 3 and common ratio 2 is:
(a) $3+2n$ (b) $3 \times 2^{n-1}$ (c) $3 \times 2^n$ (d) $2 \times 3^{n-1}$
1 markWhich of these is neither an AP nor a GP?
(a) 2, 4, 6, 8(b) 3, 9, 27, 81(c) 1, 4, 9, 16(d) 5, 5, 5, 5

Assertion (A): The sequence 2, 6, 18, 54, … is a geometric progression with common ratio 3.
Reason (R): In a GP, the ratio of any term to its immediately preceding term is always constant.

Case Study

A company offers a starting annual salary of ₹4,00,000 with a fixed annual increment of ₹25,000 every year.

1 markWhat is the salary in the 4th year?
1 markIn which year does the salary first reach ₹6,00,000?
2 marksFind the total salary earned over the first 5 years.

Answer Key

t_1=3

t_5=19

3n-2=76 \Rightarrow n=26

th term

2, 5, 14, 41

a=7,d=4

t_{15}=7+14(4)=63

a=100,d=-7

100+(n-1)(-7)=2 \Rightarrow n=15

S_{40}=820

;

S_{60}-S_{40}=1830-820=1010

Q712 to 96, step 4: 22 numbers; sum

=22 \times \frac{12+96}{2}=1188

Q8Ratio constant

=3

;

t_6=2 \times 3^5=486

r^3=8 \Rightarrow r=2

;

a=12/4=3

Q10

a=20,r=0.4

t_3=20(0.4)^2=3.2

t_n=20(0.4)^{n-1}

MCQ 1–3(b) · (b) · (c)

A/RBoth A and R are true, and R is the correct explanation of A

Case Study₹4,75,000 · 9th year · ₹22,50,000

1Sequences: Terms and Rules

Given

Find

Given

Find

2Arithmetic Progressions (AP)

Given

Find

Given

Find

3Sum of the First nnn Natural Numbers

Given

Find

4Geometric Progressions (GP)

Given

Find

Given

Find

3Sum of the First $n$ Natural Numbers