IB Maths AA HL Sl 1 3 Geometric Sequences and Series Questions

Practise IB Maths AA HL Geometric sequences and series with HL questions that develop route-specific sequence or financial reasoning and exam accuracy.

Exam points

Identify the first term and common ratio, then calculate or express the nth term of a geometric sequence.
Calculate finite geometric sums, including sigma notation and sums recovered from given terms or partial sums.
Solve inverse or threshold problems by using geometric terms or sums to find an unknown parameter or the least integer n.
Model growth, decay, repeated motion, finance or geometric patterns with a geometric sequence and interpret the result in context.
Derive or transform geometric-sequence forms in extended problems, including logarithmic, complex or recurrence-based representations when the constant ratio is the required object.

Learn this topic

Study SL 1.3—Geometric sequences and series conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 5]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

By considering $f(x)=1+x+x^{2}+\ldots+x^{n}$ as a geometric series, for $x \neq 1$ , show that $f(x)=\frac{x^{n+1}-1}{x-1}$ .

[ 2 ]

uses $S_{n}=\frac{u_{1}\left(r^{n}-1\right)}{r-1}$ with r=x and $u_{1}=1$
clear indication there are ( n+1 ) terms

f(x)=\frac{x^{n+1}-1}{x-1}

(b)

For $x \neq 1$ , show that $f_{1}(x)=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}$ .

[ 3 ]

METHOD 1

\begin{aligned} & f_{1}(x)=x f^{\prime}(x) \\ & =x \frac{(x-1)(n+1) x^{n}-1 \times\left(x^{n+1}-1\right)}{(x-1)^{2}} \end{aligned}

Note: Award M1 for attempting to use the quotient or the product rule to find $f^{\prime}(x)$ .

=x \frac{(n x+x-n-1) x^{n}-\left(x^{n+1}-1\right)}{(x-1)^{2}}\left(=x \frac{n x^{n+1}-n x^{n}-x^{n}+1}{(x-1)^{2}}\right)

Note: Award A1 for any correct manipulation of the derivative that leads to the AG.

=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}

METHOD 2
attempts to form $(x-1) f_{1}(x)$

\begin{aligned} & (x-1) f_{1}(x)=n x^{n+1}-\left(x+x^{2}+x^{3}+\ldots+x^{n}\right) \\ & f_{1}(x)=\frac{1}{x-1}\left(n x^{n+1}-\left(\frac{x^{n+1}-1}{x-1}-1\right)\right) \\ & f_{1}(x)=\frac{1}{x-1}\left(\frac{n x^{n+1}(x-1)-x^{n+1}+x}{x-1}\right)\left(=\frac{1}{x-1}\left(\frac{n x^{n+2}-n x^{n+1}-x^{n+1}+x}{x-1}\right)\right) \end{aligned}

Note: Award A1 for any correct manipulation of the derivative that leads to the AG.

f_{1}(x)=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}

All question bank results loaded

Learn this topic

IB Maths AA HL Sl 1 3 Geometric Sequences and Series Questions question 1

Question (a)

Question (b)