Recognise arithmetic and geometric progressions and understand the difference between them

Series – Recognising Arithmetic & Geometric Progressions and the Binomial Theorem

Learning Objectives

• State the Binomial Theorem for \((a+b)^n\) and recognise its limits (only non‑negative integer \(n\)).
• Write the general term \(T_{r+1}\) of a binomial expansion, locate any required term and simplify the coefficient.
• Identify whether a given sequence is an arithmetic progression (AP) or a geometric progression (GP).
• Write the correct \(n^{\text{th}}\)-term and sum formulae for APs and GPs, noting all domain restrictions.
• Explain clearly the differences between APs and GPs and apply the appropriate formula in exam‑style questions.

1. The Binomial Theorem (Syllabus 12.1)

The Binomial Theorem expands a power of a two‑term expression:

\[ (a+b)^n=\sum_{r=0}^{n}\binom{n}{r}\,a^{\,n-r}b^{\,r}, \qquad \binom{n}{r}= \frac{n!}{r!\,(n-r)!}, \] where n is a non‑negative integer.

• The expansion contains \(n+1\) terms, indexed by \(r=0,1,\dots ,n\).
• Each term consists of a binomial coefficient \(\binom{n}{r}\) multiplied by \(a^{\,n-r}b^{\,r}\).

Why does the theorem hold?

Proof by induction (outline) – The case \(n=0\) gives \((a+b)^0=1\), which matches the formula because \(\binom{0}{0}=1\). Assume the formula true for some \(n=k\). Then

\[ (a+b)^{k+1}=(a+b)(a+b)^k =(a+b)\sum_{r=0}^{k}\binom{k}{r}a^{k-r}b^{r} =\sum_{r=0}^{k}\binom{k}{r}a^{k+1-r}b^{r} +\sum_{r=0}^{k}\binom{k}{r}a^{k-r}b^{r+1}. \]

Re‑index the second sum (\(r\to r-1\)) and combine like powers of \(a\) and \(b\); the coefficient of \(a^{k+1-r}b^{r}\) becomes \(\binom{k}{r}+\binom{k}{r-1}=\binom{k+1}{r}\) (Pascal’s rule). Hence the formula holds for \(k+1\). By induction it is true for all \(n\ge0\).

Pascal’s Triangle – a visual aid

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

⋮

Row \(n\) gives the coefficients \(\binom{n}{0},\binom{n}{1},\dots ,\binom{n}{n}\) for \((a+b)^n\).

Important restriction

The syllabus does **not** require expansion for negative or fractional exponents. If a student encounters \((a+b)^{-2}\) or \((a+b)^{\frac32}\) they must state that the Binomial Theorem as given does not apply.

2. General Term of a Binomial Expansion (Syllabus 12.2)

The \((r+1)^{\text{st}}\) term (the general term) is

\[ T_{r+1}= \binom{n}{r}\,a^{\,n-r}b^{\,r},\qquad r=0,1,\dots ,n. \]

Step‑by‑step checklist for locating a required term

1. Identify the overall exponent \(n\).
2. Write the general term \(T_{r+1}= \binom{n}{r}a^{\,n-r}b^{\,r}\).
3. Set the exponent of the required variable (or the required power of a term) equal to the exponent in the general term and solve for \(r\).
4. Substitute the found value of \(r\) back into the general term.
5. Calculate the binomial coefficient and simplify the numerical factor (combine like terms if necessary).

Worked example – a non‑first/last term

Find the term containing \(x^{2}\) in \((2x-1)^{5}\).

1. \(n=5\), \(a=2x\), \(b=-1\).
2. General term: \(T_{r+1}= \binom{5}{r}(2x)^{5-r}(-1)^{r}\).
3. Power of \(x\) is \(5-r\). Set \(5-r=2\Rightarrow r=3\).
4. Substitute \(r=3\): \[ T_{4}= \binom{5}{3}(2x)^{2}(-1)^{3}=10\cdot4x^{2}\cdot(-1)= -40x^{2}. \]
5. The required term is \(-40x^{2}\).

3. What is a Progression?

A progression is an ordered list of numbers called terms. The rule that generates each term from its predecessor determines the type of progression.

3.1 Arithmetic Progression (AP) – Syllabus 12.3

• Successive terms differ by a constant called the common difference \(d\) (additive rule).
• nth‑term: \[ a_n = a_1 + (n-1)d. \]
• Sum of the first \(n\) terms (valid for \(n\ge1\)): \[ S_n = \frac{n}{2}\bigl(2a_1+(n-1)d\bigr)=\frac{n}{2}(a_1+a_n). \]

3.2 Geometric Progression (GP) – Syllabus 12.3

• Successive terms have a constant ratio called the common ratio \(r\) (multiplicative rule).
• nth‑term: \[ a_n = a_1\,r^{\,n-1}. \]
• Sum of the first \(n\) terms (for \(req1\) and \(n\ge1\)): \[ S_n = a_1\frac{1-r^{\,n}}{1-r}. \]
• If \(|r|<1\) the infinite sum exists: \[ S_{\infty}= \frac{a_1}{1-r}. \]
• When \(r=1\) the GP is actually a constant sequence; then \(S_n=n a_1\).

3.3 Quick‑diagnosis table

Sequence	Differences	Ratios	Type
4, 9, 14, 19,…	5, 5, 5	–	AP
3, 6, 12, 24,…	3, 6, 12	2, 2, 2	GP
2, 5, 10, 17,…	3, 5, 7	2.5, 2, 1.7	Neither
1, \(\frac12\), \(\frac14\), \(\frac18\),…	0.5, 0.25, 0.125	0.5, 0.5, 0.5	GP (\(|r|<1\))

Flow‑chart for identifying a progression

1. Calculate the differences of consecutive terms.
2. If all differences are equal → AP (record \(d\)).
3. If not, calculate the ratios of consecutive terms.
4. If all ratios are equal → GP (record \(r\)).
5. If neither differences nor ratios are constant → neither AP nor GP.

4. Arithmetic Progression – Formulae, Domain & Real‑World Context

• Domain restrictions: \(n\) must be a positive integer; \(d\) may be any real number (including negative, giving a decreasing AP).
• Real‑world example: Saving a fixed amount each month, e.g. deposit £50 each month → amounts form an AP with \(a_1=50\) and \(d=50\).

5. Geometric Progression – Formulae, Domain & Convergence

• Domain restrictions: \(n\) positive integer; \(req1\) for the finite‑sum formula. If \(r=1\) use \(S_n=n a_1\).
• Convergence: An infinite GP sum exists **only** when \(|r|<1\). Otherwise the series diverges.

Counter‑example for infinite sum

Sequence \(2,\,4,\,8,\,16,\dots\) has \(a_1=2\) and \(r=2\). Since \(|r|>1\), the partial sums grow without bound and no finite \(S_{\infty}\) exists.

6. Key Differences Between AP and GP

Feature	Arithmetic Progression (AP)	Geometric Progression (GP)
Defining constant	Common difference \(d\) (additive)	Common ratio \(r\) (multiplicative)
nth‑term formula	\(a_n = a_1 + (n-1)d\)	\(a_n = a_1 r^{\,n-1}\)
Sum of first \(n\) terms	\(S_n = \dfrac{n}{2}(a_1+a_n)\)	\(S_n = a_1\dfrac{1-r^{\,n}}{1-r}\) (\(req1\))
Growth pattern	Linear (or linear decay if \(d<0\))	Exponential (or exponential decay if \(|r|<1\))
Infinite sum	Never (terms do not tend to 0)	Exists only when \(|r|<1\); value \(\dfrac{a_1}{1-r}\)

7. Worked Examples

Example 1 – AP: 10th term

Sequence: \(5,\,9,\,13,\,17,\dots\)

• Differences: \(4,4,4\) → \(d=4\), \(a_1=5\).
• \(a_{10}=5+(10-1)\times4=5+36=41.\)

Example 2 – GP: Sum of first 5 terms

Sequence: \(3,\,6,\,12,\,24,\dots\)

• Ratios: \(2,2,2\) → \(r=2\), \(a_1=3\).
• \(S_5 = 3\frac{1-2^{5}}{1-2}=3\frac{1-32}{-1}=3\times31=93.\)

Example 3 – Neither AP nor GP

Sequence: \(2,\,5,\,10,\,17,\,26,\dots\)

• Differences: \(3,5,7,9\) (not constant).
• Ratios: \(2.5,2,1.7\) (not constant).
• Conclusion: not an AP nor a GP.

Example 4 – Binomial: Coefficient of \(x^{3}\) in \((x+2)^{5}\)

• General term: \(T_{r+1}= \binom{5}{r}x^{5-r}2^{r}\).
• Require exponent \(5-r=3\Rightarrow r=2\).
• Coefficient \(= \binom{5}{2}2^{2}=10\times4=40.\)

Example 5 – Infinite GP sum

Sequence: \(1,\frac12,\frac14,\frac18,\dots\)

• \(a_1=1,\; r=\frac12\) (|r|<1).
• \(S_{\infty}= \dfrac{1}{1-\frac12}=2.\)

Example 6 – Binomial: Term containing \(x^{2}\) in \((2x-1)^{5}\)

See the checklist example in Section 2 – the term is \(-40x^{2}\).

Example 7 – GP with \(|r|<1\) but negative first term

Sequence: \(-3,\;1.5,\;-0.75,\;0.375,\dots\)

• \(a_1=-3,\; r=-\tfrac12\) (|r|<1).
• \(S_{\infty}= \dfrac{-3}{1-(-\frac12)}=\dfrac{-3}{1.5}=-2.\)

8. Practice Questions

1. Determine the type of progression and write the nth‑term formula for \(7,\,11,\,15,\,19,\dots\).
2. A GP has first term \(4\) and common ratio \(3\). Find the sum of the first 6 terms.
3. Find the 8th term of an AP whose 3rd term is \(12\) and common difference is \(5\).
4. For the GP \(2,\,-6,\,18,\,-54,\dots\), state whether an infinite sum exists and, if so, calculate it.
5. Explain why the series \(1,\frac12,\frac14,\frac18,\dots\) has a finite sum, and compute that sum.
6. Expand \((3x-4)^{4}\) using the Binomial Theorem and write down the term containing \(x^{2}\). (Simplify the coefficient.)
7. Find the term independent of \(x\) in \((x^{2}+3x^{-1})^{6}\).

9. Common Mistakes to Avoid

• Confusing the common difference \(d\) with the common ratio \(r\).
• Using the AP sum formula for a GP (or vice‑versa).
• Applying the GP sum formula when \(r=1\); in that case use \(S_n=n a_1\).
• Assuming an infinite GP sum always exists – it exists only when \(|r|<1\).
• For binomial expansions, forgetting that the exponent of the first term is \(n-r\) and of the second term is \(r\).
• Neglecting to simplify the numerical coefficient after using the binomial coefficient.
• Omitting the factorial definition of \(\binom{n}{r}\) when a calculation is required.

10. Exam Tips

11. Summary

An arithmetic progression changes by a constant addition (\(d\)), producing linear growth; its nth‑term and sum formulas are based on addition. A geometric progression changes by a constant multiplication (\(r\)), producing exponential growth; its formulas involve powers of \(r\) and have a special condition for an infinite sum (\(|r|<1\)). Recognising the pattern allows you to select the correct formula quickly, a skill that is heavily tested in the Cambridge IGCSE Additional Mathematics (0606) series questions.

The Binomial Theorem expands \((a+b)^n\) for non‑negative integer \(n\). Using the general term \(T_{r+1}= \binom{n}{r}a^{\,n-r}b^{\,r}\) together with the checklist above lets you locate any required term and simplify its coefficient efficiently.

Mastery of these tools – AP/GP identification, formula application, and binomial expansion – will enable you to tackle all series‑related problems in the exam with confidence.