The sum of an arithmetic sequence

Sequences, grade 12 (4 units) Lesson 7 of 20Learning

The sum of an arithmetic sequence

Adding in pairs

We write the sum $\displaystyle 1+2+3+\ldots+100$ twice – once from the start and once from the end – and add column by column:

\begin{aligned}S&=1+2+\ldots+99+100\\S&=100+99+\ldots+2+1\end{aligned}

Each column adds up to 101, and there are 100 columns: $\displaystyle {2S}={100\cdot101}$ .
$\displaystyle {S}={\frac{100\cdot101}{2}}={5050}$ .

The sum

\displaystyle S_n

$\displaystyle S_n$ is the sum of the first $\displaystyle n$ terms: $\displaystyle {S_n}={a_1+a_2+\ldots+a_n}$ .
In an arithmetic sequence every column, as in 1 to 100, is $\displaystyle a_1+a_n$ : one step inwards adds $\displaystyle d$ on one side and takes $\displaystyle d$ away on the other. There are $\displaystyle n$ columns, so:

\displaystyle S_n

\displaystyle =

\displaystyle \frac{n(a_1+a_n)}{2}

Substituting $\displaystyle {a_n}={a_1+(n-1)d}$ gives a formula without the last term:

\displaystyle S_n

\displaystyle =

\displaystyle \frac{n(2a_1+(n-1)d)}{2}

Example: seats in a hall

A hall has 20 rows.
The first row has 12 seats, and each row has 2 seats more than the row before it.
How many seats does the hall have?

$\displaystyle {a_{20}}={12+19\cdot2}={50}$ .
$\displaystyle {S_{20}}={\frac{20(12+50)}{2}}={620}$ seats.
Or, without $\displaystyle a_{20}$ : $\displaystyle {S_{20}}={\frac{20(2\cdot12+19\cdot2)}{2}}={\frac{20\cdot62}{2}}={620}$ .

Finding

\displaystyle n

from the sum

In an arithmetic sequence $\displaystyle {a_1}={4}$ and $\displaystyle {d}={5}$ .
How many terms must be added, from the first, for the sum to be 9,090?

\frac{n(2\cdot4+(n-1)\cdot5)}{2}=9090

$\displaystyle {n(5n+3)}={18180}$ , that is $\displaystyle {5n^2+3n-18180}={0}$ .
By the quadratic formula: $\displaystyle {n}={60}$ or $\displaystyle {n}={-60.6}$ .
The number of terms is a natural number, so $\displaystyle {n}={60}$ .

Exercise 1

The arithmetic sequence: $\displaystyle 5, 9, 13, \ldots$
Calculate the sum of the first 30 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

Exercise 2

The sum: $\displaystyle 3+7+11+\ldots+199$ .

How many terms does the sum have?

Calculate the sum.

/ – 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

Exercise 3

The arithmetic sequence: $\displaystyle 2, 5, 8, \ldots$
How many terms must be added, from the first, for the sum to be 155?

/ – 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

Exercise 4

The chart shows the number of seats in the first five rows of a cinema.
The difference between the numbers of seats in neighbouring rows stays the same.
The cinema has 400 seats.
How many rows does it have?

/ – 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

Formulas: sequences, grade 12 (4 units)

Open as a page

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
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
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
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

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 the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$