Chapter 3 · Expansion & Factorisation

1.1.5 Sequences & the General Term

A sequence that jumps by the same amount every time has a formula hiding inside it — once you find that jump, you can predict the 105th term without writing out 104 of them first.

🎯 By the end of this you can…

✓Find the common difference of an arithmetic sequence
✓Build a formula Tₙ for the nth term of any sequence
✓Use that formula to jump straight to a term far down the sequence

💡 The idea

Arithmetic = same jump every time

A sequence like $7, 13, 19, 25, 31, \dots$ goes up by $6$ every single time. That fixed jump is the common difference, usually written $c . d .$ — and it's the whole key to writing a formula for any term.

🔢 Watch the pattern jump

+6→

📋 Finding the formula for the nth term

1
Find the common difference: $c . d . = T_{2} - T_{1}$ .
2
The $n$ part of the formula is just the common difference times $n$ : a trial formula $T_{n} = (c . d .) n$ .
3
Find the "term before the first term", $T_{0} = T_{1} - c . d .$ — this becomes the number added at the end.
4
Combine them: $T_{n} = (c . d .) n + T_{0}$ .

🧮 Formula for 7, 13, 19, 25, 31, … worked out

Common difference:

T_{2} - T_{1} = 13 - 7 = 6

1 / 5

✏️ Worked example

Find the formula for −4, −1, 2, 5, 8, …

1
Common difference: $- 1 - (- 4) = 3$ .
2
Term before the first: $T_{0} = - 4 - 3 = - 7$ .

✅ Answer: $T_{n} = 3 n - 7$

🔢 Watch the pattern jump

+7→

✏️ Worked example

Find the formula for 14, 11, 8, 5, 2, … (a decreasing sequence)

1
Common difference: $11 - 14 = - 3$ — negative, since the sequence is going down.
2
Term before the first: $T_{0} = 14 - (- 3) = 17$ .

✅ Answer: $T_{n} = - 3 n + 17$

💡 The idea

A geometric pattern is a sequence too

Points dividing a line into segments: 1 point makes 2 segments, 2 points make 3 segments, 3 points make 4 segments. The pattern is simply $T_{n} = n + 1$ — one more segment than points, because each new point just cuts one more piece off the line.

✏️ Worked example

49 points divide a line segment. How many segments does that make?

1
$T_{n} = n + 1$ , with $n = 49$
2
$T_{49} = 49 + 1$

✅ Answer: 50 segments

⚠️ The classic trap

The formula gives $T_{n} = 0 \cdot n + T_{0}$ only after you've simplified — never skip straight from "common difference" to writing a half-finished formula down as the final answer. Always check your formula against a term you already know, the way the worked example above checks $T_{5}$ .

🎮 Find the common difference, then the formula

⭐ 0 XP🔥 streak 0

Find

T_{n}

for

4, 6, 8, 10, 12, \dots

Find

T_{n}

for

3, 6, 9, 12, 15, \dots

Find

T_{n}

for

10, 14, 18, 22, 26, \dots

Using

T_{n} = 3 n

, find the 105th term of

3, 6, 9, \dots

Using

T_{n} = 4 n + 6

, find the 200th term of

10, 14, 18, \dots

Find

T_{n}

for

20, 17, 14, 11, \dots

If 60 points divide a line segment, how many segments are there?

How many points are needed to divide a line segment into 101 segments?

0 of 8 finished

🤔 101 points are needed to divide a line segment into how many segments, using Tₙ = n + 1?

🏅

You finished this lesson!

You can now turn any steady pattern into a formula — and use that formula to answer "what's the 200th one?" without ever writing out the first 199.

Sign In to Keep Going

You've reached the end of the free preview. Sign in — and upgrade to Plus or Pro — to keep going with Mathematics.

Full Access

Unlock all 16 lessons

Get Plus or Pro Access