Unit notes and practice questions
Everything here is for unit 1 of the Grade 12 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.
The cards filed under this chapter on the Grade 12 Mathematics page. Open one at a time.
When you use it
Use this to find the single intermediate term between two terms that forms an arithmetic sequence.
Watch out
When inserting multiple arithmetic means, find the common difference first rather than averaging the two endpoints.
Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up
When you use it
Use this when checking whether a general formula defines an arithmetic sequence.
Watch out
Testing only the first few terms by substitution is not enough. You must subtract a_n from a_{n+1} and confirm that n cancels out completely.
Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up
When you use it
Use this to calculate any term of an arithmetic sequence when the first term and common difference are known.
Watch out
Do not multiply d by n. The common difference is added n - 1 times to the first term.
Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up
When you use it
Reach for this whenever finding the common difference of a decreasing arithmetic sequence.
Watch out
Always calculate succeeding term minus preceding term. If terms decrease, the common difference is negative.
Drafted from Grade 12 Mathematics, pages 1-60, then checked twice before it went up
10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.
Question 1
Look at the sequence 2, 5, 8, 11, 14, .... What is the common difference?
Answer: B 3
Why
In an arithmetic sequence, each term is found by adding the same fixed number to the term before it. Here, 5 - 2 = 3, 8 - 5 = 3, and so on. So the common difference is 3.
Question 2
Bontu had 130 birr in January. In February she had 210 birr, and in March she had 290 birr. If she keeps saving the same amount each month, how much will she have in April?
Answer: B 370 birr
Why
Bontu adds 210 - 130 = 80 birr each month. The sequence is 130, 210, 290, 370. So in April she will have 290 + 80 = 370 birr.
Question 3
A sequence has the general term an = 2n, where n is a positive integer. What are the first five terms?
Answer: B 2, 4, 6, 8, 10
Why
Put n = 1, 2, 3, 4, 5 into the rule an = 2n. You get a1 = 2, a2 = 4, a3 = 6, a4 = 8 and a5 = 10.
Question 4
What is the sum of the first five even natural numbers?
Answer: C 30
Why
The first five even natural numbers are 2, 4, 6, 8 and 10. Their sum is 2 + 4 + 6 + 8 + 10 = 30.
Question 5
An arithmetic sequence has first term A1 = 5 and common difference d = 4. Which formula is the general term An?
Answer: B A<sub>n</sub> = 4n + 1
Why
Use the formula An = A1 + (n - 1)d. Substitute to get An = 5 + (n - 1)(4) = 5 + 4n - 4 = 4n + 1.
Question 6
Find the value of Σk=14 5k.
Answer: C 50
Why
Write out each term by putting k = 1, 2, 3, 4 into 5k. You get 5 + 10 + 15 + 20 = 50.
Question 7
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, .... Each term is the sum of the two terms before it. What is the 12th term?
Answer: C 144
Why
Keep adding the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Counting the list, the 12th term is 144.
Question 8
Consider the infinite geometric series 1/2 + 1/4 + 1/8 + .... What is its sum to infinity?
Answer: A 1
Why
The first term is G1 = 1/2 and the common ratio is r = 1/2. Since |r| is less than 1, the series converges. The sum is G1 / (1 - r) = (1/2) / (1/2) = 1.
Question 9
In an arithmetic sequence, the 3rd term is 5 and the 5th term is 11. What is the sum of the first 7 terms, S7?
Answer: B 56
Why
From A3 = A1 + 2d = 5 and A5 = A1 + 4d = 11, subtracting gives 2d = 6, so d = 3 and A1 = -1. Then A7 = -1 + 6(3) = 17. The sum is S7 = 7/2 (A1 + A7) = 7/2 (-1 + 17) = 7/2 (16) = 56.
Question 10
Find the sum of the first 99 terms of the sequence whose general term is an = 1 / (n(n + 1)). That is, find Σn=199 1 / (n(n + 1)).
Answer: B 99/100
Why
Split each term using partial fractions: 1 / (n(n + 1)) = 1/n - 1/(n + 1). When you add the terms, most of them cancel. What is left is 1 - 1/100 = 99/100.
The app holds 12 questions on this unit, marked as you go. These are the first 10.
A lesson on this chapter from YouTube. It loads only when you press play.
Sequences and series (part 2) · Watch on YouTube
Unit 1 runs from page 1 to page 60 of the Ministry of Education edition.
Ministry of Education edition, English, 426 pages
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