IB Maths AA HL/Question Bank/1.1 Number and algebra - SL content

IB Maths AA HL 1.1 Number and algebra - SL content Question Bank

HL198 questions10 previewsSyllabus linked

[Maximum number: 4]

Consider the arithmetic sequence $8,26,44, \ldots$ .

(a)

Find an expression for the $n^{\text {th }}$ term.

[ 1 ]

(b)

Write down the sum of the first n terms using sigma notation.

[ 1 ]

(c)

Calculate the sum of the first 15 terms.

[ 2 ]

[Maximum number: 7]

The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8 .

(a)

Find the first term and the common difference.

[ 4 ]

(b)

Find the smallest value of n such that the sum of the first n terms is greater than 600 .

[ 3 ]

[Maximum number: 4]

Find the sum of all the multiples of 3 between 100 and 500 .

[Maximum number: 4]

Find the value of k if $\sum_{r=1}^{\infty} k\left(\frac{1}{3}\right)^{r}=7$ .

[Maximum number: 2]

Let $f(n)=n^{5}-n, n \in \mathbb{Z}^{+}$ .

(a)

Find the values of f(3), f(4) and f(5).

[ 2 ]

[Maximum number: 6]

(a)

(i)

Find the sum of all integers, between 10 and 200, which are divisible by 7 .

(ii)

Express the above sum using sigma notation.

An arithmetic sequence has first term 1000 and common difference of -6 . The sum of the first n terms of this sequence is negative.

[ 4 ]

(b)

Find the least value of n.

[ 2 ]

[Maximum number: 5]

Given the sets A and B, use the properties of sets to prove that $A \cup\left(B^{\prime} \cup A\right)^{\prime}=A \cup B$ , justifying each step of the proof.

[Maximum number: 17]

In this question the notation $\left(a_{n} a_{n-1} \ldots a_{2} a_{1} a_{0}\right)_{b}$ is used to represent a number in base b, that has unit digit of $a_{0}$ . For example (2234) ${ }_{5}$ represents $2 \times 5^{3}+2 \times 5^{2}+3 \times 5+4=319$ and it has a unit digit of 4 .

(a)

Let x be the cube root of the base 7 number $(503231)_{7}$ .

[ 7 ]

(i)

By converting the base 7 number to base 10 , find the value of x, in base 10 .

(ii)

Express x as a base 5 number.

[ 7 ]

(b)

Let y be the base 9 number $\left(a_{n} a_{n-1} \ldots a_{1} a_{0}\right)_{9}$ . Show that y is exactly divisible by 8 if and only if the sum of its digits, $\sum_{i=0}^{n} a_{i}$ , is also exactly divisible by 8 .

[ 7 ]

(c)

Using the method from part (b), find the unit digit when the base 9 number $(321321321)_{9}$ is written as a base 8 number.

[ 3 ]

[Maximum number: 6]

The fifth term of an arithmetic sequence is equal to 6 and the sum of the first 12 terms is 45 . Find the first term and the common difference.

(a)

Use the Euclidean algorithm to show that 1463 and 389 are relatively prime.

[ 4 ]

(b)

Find positive integers a and b such that 1463 a-389 b=1.

[ 5 ]