Arithmetic and geometric sequences · Grade 11 · 5 units
In Arithmetic and geometric sequences (Grade 11 · 5 units) you practise 8 skills: The rule of a sequence: the general term and a recurrence · Arithmetic sequences: the general term · The sum of an arithmetic sequence · Geometric sequences: the general term · The sum of a geometric sequence · Infinite geometric sequences with a finite sum · Properties of sequences · A sequence on a graph. A sequence is a list of numbers in order. It can be given by a formula for the general term, which gives each term from its place, or by a recurrence, which gives each term from the one before it.
In an arithmetic sequence the difference between each term and the one before it is fixed: it is called the common difference. To reach the term in a given place, add the common difference to the first term one time fewer than the place number. The sum of the terms up to a given place is the first plus the last, times the number of terms, divided by two. In a geometric sequence each term divided by the one before it gives the same number: it is called the common ratio. To reach the term in a given place, multiply the first term by the ratio one time fewer than the place number. When the absolute value of the ratio is less than one, the sum of all the terms of the infinite sequence has a finite value: the first term divided by one minus the ratio.
Tip: Before working anything out, check whether each term is the one before plus a fixed amount, or times a fixed number.
What you learn
The rule of a sequence: the general term and a recurrence
For example: A sequence is defined by a₁ = −7 and aₙ₊₁ = aₙ + 5 . What is its fourth term?
Answer and solution
The answer: 8
One after another: a₂ = −7 + 5 = −2 , a₃ = −2 + 5 = 3 , a₄ = 3 + 5 = 8 .
Arithmetic sequences: the general term
For example: An arithmetic sequence has a₁ = 9 and d = −4 . What is a₂₀ ?
Answer and solution
The answer: −67
By the formula a₂₀ = 9 + 19 × (−4) = −67 .
The sum of an arithmetic sequence
For example: An arithmetic sequence has a₁ = −8 and d = 5 . What is S₉ , the sum of the first 9 terms?
Answer and solution
The answer: 108
The last term: a₉ = −8 + 8 × 5 = 32 . So a₁ + a₉ = 24 , and the sum is S₉ = 24 × 9 ÷ 2 = 108 .
Geometric sequences: the general term
For example: A geometric sequence has a₁ = 3 and q = 2 . What is a₆ ?
Answer and solution
The answer: 96
By the formula a₆ = 3 × 2⁵ , and 2⁵ = 32 . So a₆ = 96 .
The sum of a geometric sequence
For example: The first three terms of a geometric sequence are 1, 2, 4 . What is S₉ , the sum of the first 9 terms?
Answer and solution
The answer: 511
Here a₁ = 1 and q = 2 , so q − 1 = 1 . The sum: S₉ = 1 × (2⁹ − 1) = 1 × 511 = 511 .
Infinite geometric sequences with a finite sum
For example: An infinite geometric sequence has a₁ = −9 and q = −1 . Does the sum of all its terms have a finite value?
- Yes, since |q| < 1
- No, since |q| ≥ 1
Answer and solution
The answer: No, since |q| ≥ 1
Here |q| = 1 , which is not less than one. No, since |q| ≥ 1 : the terms do not shrink towards zero, so the sum has no finite value.
Properties of sequences
For example: The sequence aₙ = 4 + n is an arithmetic sequence. Which of these sequences is arithmetic too?
- aₙ²
- 1 ÷ aₙ
- 2^(aₙ)
- aₙ + 3
Answer and solution
The answer: aₙ + 3
In aₙ + 3 each term is 3 more than the original, so the difference between neighbours stays 1 . In aₙ² the first terms are 25, 36, 49 , and the differences 11 and 13 are not equal. In 1 ÷ aₙ the first terms are 1/5, 1/6, 1/7 , and the differences −1/30 and −1/42 are not equal. In 2^(aₙ) the first terms are 32, 64, 128 , and the differences 32 and 64 are not equal.
A sequence on a graph
For example: The graph shows the first six terms of an arithmetic sequence as the points (n, aₙ) . What is the common difference?
Answer and solution
The answer: 3
From the graph a₁ = −4 and a₂ = −1 , so d = −1 − (−4) = 3 . Each point is higher than the one before by the same amount.
Not on the site yet
Parts of this topic that are not built yet: Limits stated as “for every number, however small” · Recurring decimals as geometric sequences (enrichment) · Finding a place with logarithms (not in the programme).
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).