Review: sequences and induction

Sequences and induction, grade 11 (5 units) Lesson 23 of 23Practice

Review: sequences and induction

A sequence is defined by the recursion rule $\displaystyle {a_1}={4}$ , $\displaystyle {a_{n+1}}={\frac{2}{a_n}}$ .
Write the first four terms, and find the sum of the first 41 terms.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

In an arithmetic sequence $\displaystyle {a_6}={17}$ , and the sum of the first 10 terms is 155.
Find $\displaystyle a_1$ and $\displaystyle d$ .

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={n^2+4n}$ .
Find $\displaystyle a_n$ , and prove that the sequence is arithmetic.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The numbers $\displaystyle a$ , $\displaystyle b$ , $\displaystyle 12$ (in this order) are an arithmetic sequence, and the numbers $\displaystyle a$ , $\displaystyle b$ , $\displaystyle 16$ – a geometric sequence. $\displaystyle a$ and $\displaystyle b$ are positive.
Find $\displaystyle a$ and $\displaystyle b$ .

In a geometric sequence $\displaystyle {a_3}={20}$ and $\displaystyle {a_6}={160}$ .
Find $\displaystyle q$ , and calculate the sum of the first 8 terms.

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In an infinite geometric sequence the sum of the first two terms is 15, and the sum of the sequence is 16.
Find $\displaystyle q$ and $\displaystyle a_1$ (two solutions).

$\displaystyle a_n$ is a geometric sequence with ratio $\displaystyle q$ , and $\displaystyle {b_n}={a_{2n-1}+a_{2n}}$ .
Prove that $\displaystyle b_n$ is a geometric sequence, and find its ratio.

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A population of 2,000 deer grows by 10% every year.
In which year will it first pass 4,000 deer? Find it by trial.

$\displaystyle {1+2\cdot2+3\cdot2^2+\ldots+n\cdot2^{n-1}}={(n-1)2^n+1}$
Prove by induction that the claim holds for every natural $\displaystyle n$ .

Base case

Check the claim for $\displaystyle {n}={1}$ .

Inductive step

Write the induction hypothesis – the claim for some natural $\displaystyle k$ – and what must be proved for $\displaystyle k+1$ .

The working: start from the expression you got after putting in the hypothesis, and reach what must be proved.

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Conclusion

Prove by induction that $\displaystyle 9^n-1$ is divisible by 8 for every natural $\displaystyle n$ .

Inductive step

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Formulas: sequences and induction, grade 11 (5 units)

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A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle S_n-S_{n-1}$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n\left[2a_1+(n-1)d\right]}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of a convergent sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$

The laws of powers

A product of powers of one base
$\displaystyle a^m\cdot a^n$ $\displaystyle =$ $\displaystyle a^{m+n}$
A quotient of powers of one base
$\displaystyle \frac{a^m}{a^n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^m\right)^n$ $\displaystyle =$ $\displaystyle a^{mn}$
A power with exponent 0
$\displaystyle a^0$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A power with a negative exponent
$\displaystyle a^{-n}$ $\displaystyle =$ $\displaystyle \frac{1}{a^n}$ $\displaystyle a\neq0$

Sums proved by induction

The sum of the natural numbers
$\displaystyle 1+2+3+\ldots+n$ $\displaystyle =$ $\displaystyle \frac{n(n+1)}{2}$
The sum of the odd numbers
$\displaystyle 1+3+5+\ldots+(2n-1)$ $\displaystyle =$ $\displaystyle n^2$
The sum of the squares
$\displaystyle 1^2+2^2+3^2+\ldots+n^2$ $\displaystyle =$ $\displaystyle \frac{n(n+1)(2n+1)}{6}$
The sum of the cubes
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$