IB Mathematics AA SL - Questionbank
Topic 1 All - Number & Algebra
All Questions for Topic 1 (Number & Algebra). Sequences & Series, Exponents & Logs, Binomial Theorem, Proofs
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Question 1
[Maximum mark: 4]
Expand in descending powers of and simplify your answer.
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Question 2
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
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Question 3
[Maximum mark: 7]
Find the value of each of the following, giving your answer as an integer.
-
. [2]
-
. [2]
-
. [3]
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Question 4
[Maximum mark: 6]
Find the value of each of the following, giving your answer as an integer.
-
. [2]
-
. [2]
-
. [2]
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Question 5
[Maximum mark: 6]
Consider an arithmetic sequence
-
Find the common difference, . [2]
-
Find the th term in the sequence. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 6
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
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Question 7
[Maximum mark: 6]
Consider the infinite geometric sequence , , ,
-
Find the common ratio, . [2]
-
Find the th term. [2]
-
Find the exact sum of the infinite sequence. [2]
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Question 8
[Maximum mark: 4]
Expand in descending powers of and simplify your answer.
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Question 9
[Maximum mark: 4]
Prove that the sum of three consecutive positive integers is divisible by .
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Question 10
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]
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Question 11
[Maximum mark: 4]
Consider two consecutive positive integers, and .
Show that the difference of their squares is equal to the sum of the two integers.
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Question 12
[Maximum mark: 6]
The table shows the first four terms of three sequences: , , and .
-
State which sequence is
-
arithmetic;
-
geometric. [2]
-
-
Find the sum of the first terms of the arithmetic sequence. [2]
-
Find the exact value of the th term of the geometric sequence. [2]
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Question 13
[Maximum mark: 4]
The product of three consecutive integers is increased by the middle integer.
Prove that the result is a perfect cube.
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Question 14
[Maximum mark: 6]
Consider the expansion of .
-
Write down the number of terms in this expansion. [1]
-
Find the coefficient of the term in . [5]
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Question 15
[Maximum mark: 6]
Jeremy invests into a savings account that pays an annual interest rate of %, compounded annually.
-
Write down a formula which calculates that total value of the investment after years. [2]
-
Calculate the amount of money in the savings account after:
-
year;
-
years. [2]
-
-
Jeremy wants to use the money to put down a deposit on an apartment. Determine if Jeremy will be able to do this within a -year timeframe.[2]
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Question 16
[Maximum mark: 5]
Consider the expansion of .
-
Write down the number of terms in this expansion. [1]
-
Find the coefficient of the term in . [4]
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Question 17
[Maximum mark: 6]
Consider the infinite geometric sequence , , , , ...
-
Find the common ratio. [2]
-
Find the th term. [2]
-
Find the exact sum of the infinite sequence. [2]
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Question 18
[Maximum mark: 6]
Let , , . Write down the following expressions in terms of , and .
-
[2]
-
[2]
-
[2]
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Question 19
[Maximum mark: 7]
An arithmetic sequence is given by , ,
-
Write down the value of the common difference, . [1]
-
Find
-
;
-
. [4]
-
-
Given that , find the value of . [2]
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Question 20
[Maximum mark: 7]
Let and . Write down the following expressions in terms of and .
-
[2]
-
[2]
-
[3]
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Question 21
[Maximum mark: 6]
Consider the following sequence of figures.
Figure 1 contains line segments.
-
Given that Figure contains line segments, show that .[3]
-
Find the total number of line segments in the first figures. [3]
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Question 22
[Maximum mark: 6]
Hannah buys a car for . The value of the car depreciates by % each year.
-
Find the value of the car after years. [3]
Patrick buys a car for . The car depreciates by a fixed percentage each year, and after years it is worth .
- Find the annual rate of depreciation of the car.
[3]
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Question 23
[Maximum mark: 6]
A D printer builds a set of Eifel Tower Replicas in different sizes. The height of the largest tower in this set is cm. The heights of successive smaller towers are % of the preceding larger tower, as shown in the diagram below.
-
Find the height of the smallest tower in this set. [3]
-
Find the total height if all towers were placed one on top of another. [3]
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Question 24
[Maximum mark: 6]
Let , where . Write down each of the following
expressions
in terms of .
-
[2]
-
[2]
-
[2]
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Question 25
[Maximum mark: 6]
Julia wants to buy a house that requires a deposit of Australian dollars (AUD).
Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of AUD Julia needs to invest to reach AUD after years. Give your answer correct to the nearest dollar. [3]
Julia's parents offer to add AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of %, compounded quarterly.
- Find the number of years it would take Julia to save the
AUD if she accepts her parents money and
follows their advice. Give your answer correct to the nearest
year. [3]
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Question 26
[Maximum mark: 6]
The fifth term, , of a geometric sequence is . The sixth term, , is .
-
Find the common ratio of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Calculate the sum of the first terms of the sequence. [2]
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Question 27
[Maximum mark: 6]
In an arithmetic sequence, , .
-
Find the common difference. [2]
-
Find the first term. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 28
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Mia deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded semi-annually.
-
Find the amount of interest that Mia will earn over the next years. [3]
Ella also deposits AUD into a bank account. Her bank pays a nominal annual rate of %, compounded monthly. In years, the total amount in Ella's account will be AUD.
- Find the amount that Ella deposits in the bank account.
[3]
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Question 29
[Maximum mark: 5]
Maria invests into a savings account that pays a nominal annual interest rate of %, compounded monthly.
-
Calculate the amount of money in the savings account after years. [2]
-
Calculate the number of years it takes for the account to reach . [3]
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Question 30
[Maximum mark: 6]
The first three terms of a geometric sequence are , , .
-
Find the value of the common ratio, . [2]
-
Find . [2]
-
Find . [2]
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Question 31
[Maximum mark: 6]
In an arithmetic sequence, , .
-
Find the common difference. [2]
-
Find the first term. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 32
[Maximum mark: 6]
Emily deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of money that Emily will have in her bank account after years. Give your answer correct to two decimal places. [3]
Emily will withdraw the money back from her bank account when the amount reaches AUD.
- Find the time, in months, until Emily withdraws the
money from her bank account. [3]
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Question 33
[Maximum mark: 6]
-
Show that for . [3]
-
Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
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Question 34
[Maximum mark: 6]
A tennis ball bounces on the ground times. The heights of the bounces, form a geometric sequence. The height that the ball bounces the first time, , is cm, and the second time, , is cm.
-
Find the value of the common ratio for the sequence. [2]
-
Find the height that the ball bounces the tenth time, . [2]
-
Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to decimal places. [2]
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Question 35
[Maximum mark: 5]
In an arithmetic sequence, the sum of the 2nd and 6th term is .
Given that the sum of the first six terms is , determine the first
term and common difference of the sequence.
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Question 36
[Maximum mark: 6]
Ali bought a car for . The value of the car depreciates by % each year.
-
Find the value of the car at the end of the first year. [2]
-
Find the value of the car after years. [2]
-
Calculate the number of years it will take for the car to be worth exactly half its original value. [2]
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