Strand 4 — Algebra

Sequences and Series

5th Year · 6th Year (Leaving Cert)

✓By the end of this lesson students will be able to define and identify arithmetic and geometric sequences.
✓By the end of this lesson students will be able to calculate the general term (n-th term) of arithmetic and geometric sequences.
✓By the end of this lesson students will be able to calculate the sum of the first n terms of arithmetic and geometric series.
✓By the end of this lesson students will be able to determine if a geometric series converges and calculate its sum to infinity.
✓By the end of this lesson students will be able to apply sequences and series to solve real-world problems, including financial applications (HL).

Key concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

T_n = a + (n-1)d

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence.

S_n = n/2 [2a + (n-1)d] or S_n = n/2 (a + T_n)

Geometric Sequence

A geometric sequence is a sequence of numbers such that the ratio of consecutive terms is constant. This constant ratio is called the common ratio, denoted by 'r'.

T_n = ar^(n-1)

Geometric Series

A geometric series is the sum of the terms of a geometric sequence.

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 absolute value of the common ratio 'r' must be less than 1 (i.e., -1 < r < 1). If this condition is met, the sum approaches a finite value as the number of terms approaches infinity.

S_∞ = a / (1 - r) (for |r| < 1)

Key facts to remember

1The general term of an arithmetic sequence is T_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference.
2The sum of the first n terms of an arithmetic series is S_n = n/2 [2a + (n-1)d].
3The general term of a geometric sequence is T_n = ar^(n-1), where 'a' is the first term and 'r' is the common ratio.
4The sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r) (for r ≠ 1).
5The sum to infinity of a geometric series is S_∞ = a / (1 - r), valid only if the common ratio |r| < 1.
6The common difference 'd' can be found by subtracting any term from its succeeding term (T_n - T_(n-1)).
7The common ratio 'r' can be found by dividing any term by its preceding term (T_n / T_(n-1)).

Worked examples

Example 1

An arithmetic sequence has a first term of 3 and a common difference of 4. Find the 10th term and the sum of the first 10 terms.

IIdentify the given values: a = 3, d = 4, n = 10.

IITo find the 10th term (T_10), use the formula for the general term of an arithmetic sequence: T_n = a + (n-1)d.

IIISubstitute the values: T_10 = 3 + (10-1)4

IVT_10 = 3 + (9)4

VT_10 = 3 + 36

VIT_10 = 39.

VIITo find the sum of the first 10 terms (S_10), use the formula for the sum of an arithmetic series: S_n = n/2 [2a + (n-1)d].

VIIISubstitute the values: S_10 = 10/2 [2(3) + (10-1)4]

9S_10 = 5 [6 + (9)4]

10S_10 = 5 [6 + 36]

11S_10 = 5 [42]

12S_10 = 210.

Answer

The 10th term is 39. The sum of the first 10 terms is 210.

Example 2

A geometric sequence has a first term of 2 and a common ratio of 3. Find the 5th term and the sum of the first 5 terms.

IIdentify the given values: a = 2, r = 3, n = 5.

IITo find the 5th term (T_5), use the formula for the general term of a geometric sequence: T_n = ar^(n-1).

IIISubstitute the values: T_5 = 2(3)^(5-1)

IVT_5 = 2(3)^4

VT_5 = 2(81)

VIT_5 = 162.

VIITo find the sum of the first 5 terms (S_5), use the formula for the sum of a geometric series: S_n = a(1 - r^n) / (1 - r).

VIIISubstitute the values: S_5 = 2(1 - 3^5) / (1 - 3)

9S_5 = 2(1 - 243) / (-2)

10S_5 = 2(-242) / (-2)

11S_5 = -484 / -2

12S_5 = 242.

Answer

The 5th term is 162. The sum of the first 5 terms is 242.

Example 3

A ball is dropped from a height of 10 metres. After each bounce, it rises to 80% of its previous height. (i) What height does it reach after the 3rd bounce? (ii) What is the total vertical distance travelled by the ball before it comes to rest? (HL)

I(i) Calculate the height after each bounce:

IIInitial drop height = 10 m.

IIIHeight after 1st bounce (T_1) = 10 × 0.8 = 8 m.

IVHeight after 2nd bounce (T_2) = 8 × 0.8 = 6.4 m.

VHeight after 3rd bounce (T_3) = 6.4 × 0.8 = 5.12 m.

VIAlternatively, using T_n = ar^(n-1) where 'a' is the first rise height: a = 8, r = 0.8. The height after the 3rd bounce is T_3 for the *rise* sequence.

VIIT_3 = 8 × (0.8)^(3-1) = 8 × (0.8)^2 = 8 × 0.64 = 5.12 m.

VIII(ii) Calculate the total vertical distance.

9The total distance consists of the initial drop, and then the sum of all subsequent rises and falls.

10Initial drop = 10 m.

11The sequence of heights the ball *rises* to is a geometric sequence: 8, 6.4, 5.12, ...

12For this sequence: a = 8, r = 0.8.

13Since |r| = 0.8 < 1, the sum to infinity exists for the rises.

14Sum of all rises (S_∞_rise) = a / (1 - r) = 8 / (1 - 0.8) = 8 / 0.2 = 40 m.

15The sequence of heights the ball *falls* after the initial drop is identical to the sequence of rises: 8, 6.4, 5.12, ...

16Sum of all falls (S_∞_fall) = 40 m.

17Total vertical distance = Initial drop + Sum of all rises + Sum of all falls

18Total vertical distance = 10 + 40 + 40 = 90 m.

Answer

(i) The height it reaches after the 3rd bounce is 5.12 metres. (ii) The total vertical distance travelled by the ball before it comes to rest is 90 metres.

For part (ii), ensure you account for both the upward and downward journeys after the initial drop.

Common mistakes

✗Confusing the formulas for arithmetic and geometric sequences/series.
✗Incorrectly identifying the first term 'a' or the common difference 'd' / common ratio 'r' from the problem description.
✗Forgetting the crucial condition |r| < 1 for the sum to infinity of a geometric series to exist.
✗Errors in algebraic manipulation, especially when dealing with negative common ratios or large powers.
✗In application problems, not correctly setting up the series (e.g., confusing initial value with the first term of the relevant sequence, or missing parts of the total sum like initial drops/rises).

Exam tips

★Always write down the relevant formula before substituting values. This helps you earn partial marks even if you make a calculation error.
★Clearly identify and list the values of 'a', 'd' (or 'r'), and 'n' from the problem statement at the start of your solution.
★For sum to infinity questions, always check and state that |r| < 1 before applying the formula.
★For application problems, drawing a simple diagram or listing the first few terms can help you understand the sequence and avoid errors in setting up 'a' and 'r'.
★Pay close attention to the wording of the question: distinguish between finding a specific term (T_n) and finding a sum (S_n).

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum