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Analysis and Approaches SL — Questionbank

All Concepts - Number & Algebra

All questions in Topic 1 Number & Algebra

Question 21

NO CALCULATOR

Medium

[Maximum mark: 7]

The sum of the first n terms of an arithmetic sequence is given by $S_{n} = rn^{2} + sn$ , where r and s are constants. It is given that $S_{3} = 24$ and $S_{4} = 40$


(a) Show that r = 2 and s = 6	[5]
(b) Find the value of the 4th term, $u_{4}$	[2]

Question 22

NO CALCULATOR

Mild

[Maximum mark: 5]

The n^th term of an arithmetic sequence is given by $u_{n} = 20 - 4n$


(a) State the value of the first term, $u_{1}$	[1]
(b) Find the value of n when the n^th term of the sequence is −28	[2]
(c) Write down the common difference of the sequence.	[2]

Question 23

NO CALCULATOR

Spicy

[Maximum mark: 13]

Consider the function ${f(x) = a}^{x}$ , $where\ x,a \in R,\ x > 0,\ a > 1$

The graph of $f(x)$ contains the point (3,81)


(a) Show that a = 3	[2]
(b) Write down an expression for f⁻¹(x)	[1]
(c) Find the value of f⁻¹(243) Consider the arithmetic sequence log₃16, log₃p , log₃q , log₃128 , where p > 1 and q > 1	[2]
(d.i) Show that 16, p, q, 128 are four consecutive terms in a geometric sequence.	[4]
(d.ii) Find the value of p and the value of q.	[4]

Question 24

NO CALCULATOR

Medium

[Maximum mark: 12]

Consider the series $\ln x\mathbf{+}q\ln x\mathbf{+}\frac{1}{2}\ln x\mathbf{+}\ldots$ $where\ x \in R,\ x > 1,\ q \in R,\ q \neq 0$


(a) Geometric Case (a.i) Show that $q\mathbf{=} \pm \frac{1}{4}$	[2]
(a.ii) Given that q > 0 and the sum to infinity of the series is 4, find the value of x.	[3]
(b) Arithmetic Case Now consider the case where the series is arithmetic with common difference d. (b.i) Show that $q = \frac{5}{8}$	[3]
(b.ii) Write down d in the form k lnx, where k∈Q.	[1]
(b.iii) The sum of the first n terms of the series is ln x Find the value of n	[3]

Question 25

CALCULATOR

Medium

[Maximum mark: 7]

Sofia buys a computer for $2400. The value of the computer decreases by 20% in the first year.

(a) Find the value of the computer at the end of the first year.

After the first year, the value of the computer decreases by 12% in each subsequent year.

[2]

(b) Find the value of Sofia’s computer 8 years after she buys it, giving your answer to the nearest dollar.

When Sofia has owned the computer for n complete years, the value of the computer is less than 5% of its original value.

[2]

(c) Find the least value of n.

[3]

Question 26

CALCULATOR

Medium

[Maximum mark: 15]

Maria and Adrian have each recently received some money. Maria received a scholarship and Adrian received a bonus.

Maria had two options to choose from to receive her scholarship. In both options she receives a payment on the first day of each month for four years.

Option A: Each payment is $3500.
Option B: The first payment is $1800. In each month which follows, the payment is 5% more than the previous month.


(a.i) Find the total amount Maria would receive if she chooses Option A.	[1]
(a.ii) Find the total amount Maria would receive if she chooses Option B. Adrian received a bonus of $200 000. He invested his bonus in an account that pays a nominal annual interest rate of 4.8%, compounded monthly. The interest is added on the last day of each month.	[3]
(b) Write down an expression for the value of Adrian’s investment after nnn years. Maria chose Option B and received her first payment on 1st January 2024. Adrian invested his bonus on the same day.	[1]
(c.i) Find the total value of Maria’s scholarship payments and Adrian’s investment on the last day of the 9th month.	[3]
(c.ii) Find the minimum number of complete months before the combined value of Maria’s scholarship payments and Adrian’s investment is at least $320 000. At the end of the four years, Maria invests $40 000 for a further five years in a second account that pays a nominal interest rate of r%r\%r% per annum, compounded quarterly.	[4]
(d) Find the value of rrr if this investment grows to $60 000 after five years.	[3]