The first term and the difference

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

The first term and the difference

Two facts, two equations

Every term of an arithmetic sequence is fixed by $\displaystyle a_1$ and $\displaystyle d$ . So two facts about the sequence give a system of two equations.

In an arithmetic sequence $\displaystyle {a_4}={17}$ and $\displaystyle {a_{11}}={45}$ :

$\displaystyle {a_1+3d}={17}$ and $\displaystyle {a_1+10d}={45}$ .
Subtract the first equation from the second: $\displaystyle {7d}={28}$ , so $\displaystyle {d}={4}$ .
$\displaystyle {a_1}={17-3\cdot4}={5}$ .

A system with a quadratic equation

In an arithmetic sequence $\displaystyle {a_3}={7}$ , and the product $\displaystyle a_2\cdot a_5$ is 55.

Write the terms with $\displaystyle a_3$ : $\displaystyle {a_2}={7-d}$ and $\displaystyle {a_5}={7+2d}$ .
$\displaystyle {(7-d)(7+2d)}={55}$ , that is $\displaystyle {49+7d-2d^2}={55}$ , so $\displaystyle {2d^2-7d+6}={0}$ .
$\displaystyle {d}={2}$ or $\displaystyle {d}={1.5}$ .
Each solution is a sequence of its own: $\displaystyle {d}={2}$ gives $\displaystyle {a_1}={7-2\cdot2}={3}$ , and $\displaystyle {d}={1.5}$ gives $\displaystyle {a_1}={4}$ .
A check: $\displaystyle {5\cdot11}={55}$ and also $\displaystyle {5.5\cdot10}={55}$ .

Three neighbouring terms

Three neighbouring terms of an arithmetic sequence are handy to write as $\displaystyle a-d$ , $\displaystyle a$ , $\displaystyle a+d$ : in their sum $\displaystyle d$ cancels.

The sum of three neighbouring terms of an arithmetic sequence is 15, and their product is 80.

$\displaystyle {(a-d)+a+(a+d)}={3a}={15}$ , so $\displaystyle {a}={5}$ .
$\displaystyle {(5-d)\cdot5\cdot(5+d)}={80}$ , that is $\displaystyle {25-d^2}={16}$ , so $\displaystyle {d}={3}$ or $\displaystyle {d}={-3}$ .
The numbers: $\displaystyle 2, 5, 8$ or $\displaystyle 8, 5, 2$ .

Exercise 1

In an arithmetic sequence $\displaystyle {a_5}={23}$ and $\displaystyle {a_{12}}={58}$ .
Find $\displaystyle a_1$ and $\displaystyle d$ .

Exercise 2

In an arithmetic sequence $\displaystyle {a_7}={3a_2}$ and $\displaystyle {a_{10}}={42}$ .
Write a system of equations in $\displaystyle a_1$ and $\displaystyle d$ , and solve it.

Exercise 3

In an arithmetic sequence $\displaystyle {a_2+a_6}={26}$ and $\displaystyle {a_3\cdot a_5}={160}$ .
Find $\displaystyle a_1$ and $\displaystyle d$ . (Two solutions.)

Exercise 4

The sum of three neighbouring terms of an arithmetic sequence is 21, and the sum of their squares is 155.
Call the terms $\displaystyle 7-d$ , $\displaystyle 7$ , $\displaystyle 7+d$ . Find $\displaystyle d$ and the numbers.

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
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$