Pure mathematics

Sequences & Series

Year 12 · Year 13

✓By the end of this lesson students will be able to understand and use sigma notation for sums of series.
✓By the end of this lesson students will be able to work with arithmetic sequences and series, including finding the nth term and the sum of the first n terms.
✓By the end of this lesson students will be able to work with geometric sequences and series, including finding the nth term, the sum of the first n terms, and the sum to infinity.
✓By the end of this lesson students will be able to define and use recurrence relations to generate terms of a sequence.
✓By the end of this lesson students will be able to apply the binomial theorem to expand expressions for positive integer powers and for non-integer or negative powers.

Key concepts

Sequences

A sequence is an ordered list of numbers. Each number in the sequence is called a term. Sequences can be finite (have a limited number of terms) or infinite (continue indefinitely). Terms are often denoted by a_n, where n is the term number.

Series and Sigma Notation

A series is the sum of the terms in a sequence. Sigma notation (Σ) is a concise way to represent the sum of a series. The expression Σ_{i=k}^m f(i) means to sum the values of f(i) for integer values of i from k to m inclusive.

Arithmetic Sequences

An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d. The first term is denoted by a.

a_n = a + (n-1)d (nth term)

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms is denoted by S_n.

S_n = n/2 (2a + (n-1)d) or S_n = n/2 (a + l) (where l is the last term)

Geometric Sequences

A geometric sequence (or geometric progression) is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted by r. The first term is denoted by a.

a_n = ar^(n-1) (nth term)

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. The sum of the first n terms is denoted by S_n.

S_n = a(1 - r^n) / (1 - r) (for r ≠ 1)

Sum to Infinity of a Geometric Series

For a geometric series to have a finite sum to infinity, the common ratio r must satisfy the condition |r| < 1. If this condition is met, the sum to infinity, S_∞, can be calculated.

S_∞ = a / (1 - r) (for |r| < 1)

Recurrence Relations

A recurrence relation defines each term of a sequence based on one or more preceding terms. It requires an initial term (or terms) to start the sequence. For example, a_{n+1} = f(a_n).

Binomial Expansion (Positive Integer Powers)

The binomial theorem provides a formula for expanding expressions of the form (a+b)^n where n is a positive integer. The coefficients are given by binomial coefficients, (n choose k), often written as ^nC_k or (n k).

(a+b)^n = Σ_{k=0}^n (n choose k) a^(n-k) b^k where (n choose k) = n! / (k!(n-k)!)

Binomial Expansion (Non-Integer or Negative Powers)

The binomial expansion can also be used for expressions of the form (1+x)^n where n is not a positive integer (i.e., n is a rational number, negative integer, or fraction). This expansion is valid only for |x| < 1.

(1+x)^n = 1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 + ... (valid for |x| < 1)

Key facts to remember

1The nth term of an arithmetic sequence is a_n = a + (n-1)d.
2The sum of the first n terms of an an arithmetic series is S_n = n/2 (2a + (n-1)d) or S_n = n/2 (a + l).
3The nth term of a geometric sequence is a_n = ar^(n-1).
4The sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r).
5The sum to infinity of a geometric series is S_∞ = a / (1 - r), valid only when |r| < 1.
6The binomial expansion for a positive integer n is (a+b)^n = Σ_{k=0}^n (n choose k) a^(n-k) b^k.
7The binomial expansion for (1+x)^n where n is not a positive integer is 1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 + ..., valid for |x| < 1.
8A recurrence relation defines terms of a sequence based on previous terms, e.g., a_{n+1} = f(a_n).

Worked examples

Example 1

Find the sum of the series Σ_{k=1}^{20} (3k - 1).

IIdentify the first term a and the common difference d. For k=1, a_1 = 3(1) - 1 = 2. For k=2, a_2 = 3(2) - 1 = 5. So, a = 2 and d = a_2 - a_1 = 5 - 2 = 3.

IIIdentify the number of terms n. The sum goes from k=1 to k=20, so n = 20.

IIIUse the formula for the sum of an arithmetic series: S_n = n/2 (2a + (n-1)d).

IVSubstitute the values: S_{20} = 20/2 (2(2) + (20-1)3).

VS_{20} = 10 (4 + 19 × 3).

VIS_{20} = 10 (4 + 57).

VIIS_{20} = 10 (61).

Answer

S_{20} = 610

This series is arithmetic because the general term 3k-1 is a linear function of k.

Example 2

A geometric sequence has first term 100 and common ratio 0.8. Find the sum of the first 10 terms and the sum to infinity.

IIdentify a and r: a = 100, r = 0.8.

IIFor the sum of the first 10 terms, n = 10. Use the formula S_n = a(1 - r^n) / (1 - r).

IIIS_{10} = 100(1 - 0.8^{10}) / (1 - 0.8).

IVS_{10} = 100(1 - 0.1073741824) / 0.2.

VS_{10} = 100(0.8926258176) / 0.2.

VIS_{10} = 89.26258176 / 0.2.

VIIS_{10} = 446.3129088.

VIIIFor the sum to infinity, check the condition |r| < 1. Here, |0.8| < 1, so a sum to infinity exists.

9Use the formula S_∞ = a / (1 - r).

10S_∞ = 100 / (1 - 0.8).

11S_∞ = 100 / 0.2.

Answer

S_{10} = 446.31 (to 2 decimal places), S_∞ = 500

Always check the condition |r| < 1 before calculating the sum to infinity.

Example 3

Find the first four terms in the expansion of (1 - 2x)^(1/2) in ascending powers of x, stating the range of values of x for which the expansion is valid.

IIdentify n = 1/2 and X in the (1+X)^n form. Here, X = -2x.

IIUse the binomial expansion formula: (1+X)^n = 1 + nX + n(n-1)/2! X^2 + n(n-1)(n-2)/3! X^3 + ...

IIISubstitute n = 1/2 and X = -2x:

IVTerm 1: 1

VTerm 2: nX = (1/2)(-2x) = -x

VITerm 3: n(n-1)/2! X^2 = (1/2)(1/2 - 1)/2! (-2x)^2 = (1/2)(-1/2)/2 (4x^2) = (-1/4)/2 (4x^2) = -1/8 (4x^2) = -1/2 x^2

VIITerm 4: n(n-1)(n-2)/3! X^3 = (1/2)(1/2 - 1)(1/2 - 2)/3! (-2x)^3 = (1/2)(-1/2)(-3/2)/6 (-8x^3) = (3/8)/6 (-8x^3) = 3/48 (-8x^3) = 1/16 (-8x^3) = -1/2 x^3

VIIICombine the terms: 1 - x - 1/2 x^2 - 1/2 x^3 + ...

9Determine the range of validity: The expansion (1+X)^n is valid for |X| < 1. So, |-2x| < 1.

102|x| < 1 implies |x| < 1/2.

Answer

1 - x - 1/2 x^2 - 1/2 x^3 + ... for |x| < 1/2

Be careful with signs and fractions when substituting into the binomial expansion formula.

Common mistakes

✗Confusing the formulas for arithmetic and geometric sequences/series, especially d vs r.
✗Incorrectly identifying the first term a or the number of terms n from sigma notation or a problem description.
✗Forgetting to check the condition |r| < 1 before calculating the sum to infinity of a geometric series.
✗Errors in calculating binomial coefficients or signs, particularly when n is negative or fractional, or when x is negative in (1+x)^n.
✗Not stating the range of validity for binomial expansions of non-integer or negative powers, or incorrectly determining it (e.g., |ax| < 1 not |x| < 1).

Exam tips

★Always write down the relevant formula before substituting values; this helps avoid errors and can earn method marks.
★Clearly identify the values of a, d, r, and n from the problem statement before starting calculations.
★For binomial expansions, especially with non-integer powers, be meticulous with fractions and negative signs in the coefficients.
★When asked for a range of validity for binomial expansions, ensure you apply the condition |X| < 1 correctly to the 'X' part of (1+X)^n.

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