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Identifying Arithmetic Progressions

If you save £5 every single week, your total savings grow in a perfectly predictable pattern. This is an example of an arithmetic progression, frequently referred to as a linear sequence in Edexcel papers. The defining feature is that the difference between any two consecutive terms is always constant.

This constant value added or subtracted is the common difference ( $d$ ). You can identify if a sequence is arithmetic by checking if the difference between terms is consistent (i.e., $Term_2 - Term_1 = Term_3 - Term_2$ ). Unlike geometric sequences, arithmetic progressions do not multiply terms by a constant factor; they only ever add or subtract.

The General Formula for the nth Term

The nth term is an algebraic rule used to find the value of a term at any positive integer position $n$ . The first term ( $a$ ) is the starting value of the sequence when $n = 1$ . The general formula must be memorised as it is not provided on the Edexcel formula sheet:

u_n = a + (n - 1)d

Worked Example: Algebraic Simplification

Find the nth term of the sequence $7, 12, 17, 22 \dots$

Step 1: Identify the first term ( $a$ ) and common difference ( $d$ ). $a = 7$ , $d = 12 - 7 = 5$

Step 2: Substitute these values into the general formula. $u_n = 7 + (n - 1)5$

Step 3: Expand the brackets. $u_n = 7 + 5n - 5$

Step 4: Collect like terms to simplify. $u_n = 5n + 2$

Worked Example: Checking a Term

Is $99$ in the sequence $15, 21, 27 \dots$ ?

Step 1: Find the nth term rule using the zero term method. The common difference is $d = 6$ . The "zero term" (one step back from $15$ ) is $15 - 6 = 9$ . The sequence rule is $6n + 9$ .

Step 2: Set the rule equal to the target number. $6n + 9 = 99$

Step 3: Solve for $n$ . $6n = 90$ $n = 15$

Step 4: Interpret the result. Since $n$ is an integer, $99$ is exactly the 15th term in the sequence. If $n$ had been a decimal, the number would not be in the sequence.

Identifying Geometric Progressions

A linear sequence adds a constant amount each time, but a geometric progression (GP) multiplies, leading to rapid explosions in size or rapid decay. Each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio ( $r$ ).

You can find $r$ by dividing any term by the previous term ( $r = \frac{u_{n+1}}{u_n}$ ). If , the sequence shows , and if , it shows . If is a negative number, the sequence does not just grow or shrink; the signs of the terms will alternate between positive and negative.

Calculating with Geometric Progressions

The Edexcel specification often represents these sequences simply as $r^n$ , but the formal general formula used to find the nth term of a geometric progression is:

u_n = ar^{n-1}

Higher tier students must be able to handle common ratios that are fractions or a surd (an irrational number left exactly inside a root sign). When substituting fractions into the formula, it is essential to use brackets around the fraction before applying the power.

Worked Example: Finding the nth Term Expression

Find the nth term expression for a geometric progression where the second term is $-320$ and the common ratio is $-\frac{2}{5}$ .

Step 1: Find the first term ( $a$ ) using the formula $u_2 = ar$ . $-320 = a \times (-\frac{2}{5})$

Step 2: State the general nth term expression using $u_n = ar^{n-1}$ . $u_n = 800 \times (-\frac{2}{5})^{n-1}$

Worked Example: Fractional Ratio

Find the 6th term of the sequence $80, 40, 20, 10 \dots$

Step 1: Find the common ratio ( $r$ ) and first term ( $a$ ). $r = \frac{40}{80} = \frac{1}{2}$

Step 2: Substitute into the formula for $n=6$ . $u_6 = 80 \times (\frac{1}{2})^{6-1}$

Step 3: Calculate the power. $u_6 = 80 \times (\frac{1}{2})^5$

Step 4: Find the final answer. $u_6 = \frac{80}{32} = 2.5$

Worked Example: Generating Terms with a Surd Ratio

Find the 5th term of the sequence where the nth term rule is $(\sqrt{3})^n$ .

Step 1: Substitute $n=5$ into the expression. $u_5 = (\sqrt{3})^5$

Step 2: Expand the surd multiplication. $u_5 = \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

Step 3: Simplify the pairs of identical surds ( $\sqrt{3} \times \sqrt{3} = 3$ ).

Step 4: Calculate the final exact form. $u_5 = 9\sqrt{3}$

Worked Example: Finding a Surd Common Ratio (Higher Tier)

Find the common ratio ( $r$ ) of a geometric progression where the first term is $3 + 2\sqrt{2}$ and the second term is $13 + 9\sqrt{2}$ .

Step 1: Use the ratio formula $r = \frac{u_2}{u_1}$ .

Step 2: Rationalise the denominator by multiplying the numerator and denominator by the conjugate $(3 - 2\sqrt{2})$ . $r = \frac{(13 + 9 \sqrt{2}) (3 - 2 \sqrt{2})}{(3}$

Step 3: Expand the brackets for the numerator and denominator. Numerator: $39 - 26\sqrt{2} + 27\sqrt{2} - 36 = 3 + \sqrt{2}$ Denominator:

Step 4: Simplify to find the exact form of $r$ . $r = 3 + \sqrt{2}$

Students often use $n=0$ to try and find the first term of a sequence, but the position variable $n$ must always be a positive integer starting at $n=1$ .

In 'deduce the nth term' questions, examiners usually award one mark for correctly identifying the common difference (e.g., writing $4n$ ) and a second mark for finding the correct constant.

When a linear sequence is decreasing (e.g., $20, 15, 10 \dots$ ), ensure you include the negative sign in your nth term expression (write $-5n$ , not $5n$ ).

For geometric progressions, remember the power in the general formula is $n-1$ , not $n$ . Always verify your formula works by checking it against the first term.

If calculating a geometric sequence where the common ratio is a fraction or surd, Edexcel mark schemes frequently require the final answer to be left in exact form rather than a rounded decimal.

When finding the common ratio of a geometric progression containing surds (Higher Tier), you will often need to rationalise the denominator by multiplying the numerator and denominator by the conjugate.

A sequence where the difference between any two consecutive terms is always constant.

An alternative name for an arithmetic progression commonly used in Edexcel exam papers.

The constant value added to, or subtracted from, each successive term in an arithmetic progression.

A general algebraic rule or formula used to find the value of a sequence's term at any specific position.

The starting value of a sequence, represented algebraically as 'a' or 'u₁'.

A sequence where each term is found by multiplying the previous term by a constant, non-zero number.

The constant multiplier between successive terms in a geometric progression.

An irrational number expressed exactly using a root sign to avoid rounding decimals.

A process where a quantity increases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where |r| > 1.

A process where a quantity decreases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where 0 < |r| < 1.

The process of removing a surd or irrational number from the denominator of a fraction.

A binomial formed by negating the second term of a binomial, often used to rationalise denominators containing surds.