The sum of a geometric sequence

Sequences, grade 12 (4 units) Lesson 14 of 20Learning

The sum of a geometric sequence

The sum formula

To add $\displaystyle {S_n}={a_1+a_1q+\ldots+a_1q^{n-1}}$ , we multiply the whole sum by $\displaystyle q$ : each term moves to the next place.

\begin{aligned}qS_n&=a_1q+a_1q^2+\ldots+a_1q^{n}\\S_n&=a_1+a_1q+\ldots+a_1q^{n-1}\end{aligned}

Subtract: all the terms in the middle cancel, leaving $\displaystyle {qS_n-S_n}={a_1q^n-a_1}$ .
$\displaystyle {S_n(q-1)}={a_1(q^n-1)}$ , so:

\displaystyle S_n

\displaystyle =

\displaystyle \frac{a_1(q^n-1)}{q-1}

\displaystyle q\neq1

When $\displaystyle {q}={1}$ all the terms are equal, and $\displaystyle {S_n}={n\cdot a_1}$ .

Example: grains of rice

One grain of rice is put on the first square of a board, 2 on the second, 4 on the third – on each square twice as many as on the square before.
How many grains are on the first 10 squares?

$\displaystyle {a_1}={1}$ , $\displaystyle {q}={2}$ , $\displaystyle {n}={10}$ .
$\displaystyle {S_{10}}={\frac{1\cdot(2^{10}-1)}{2-1}}={1023}$ .

A negative ratio

The formula holds when $\displaystyle q$ is negative too. In the sequence $\displaystyle 2, -6, 18, \ldots$ ( $\displaystyle {q}={-3}$ ):

S_5=\frac{2((-3)^5-1)}{-3-1}=\frac{2\cdot(-244)}{-4}=122

Check: $\displaystyle {2-6+18-54+162}={122}$ .

Sub-sequences of a geometric sequence

The terms in the odd places, $\displaystyle a_1, a_3, a_5, \ldots$ : between any two we multiply by $\displaystyle q$ twice – a geometric sequence with ratio $\displaystyle q^2$ . So too in the even places, with $\displaystyle a_2$ first.
Change the sign of the terms in the even places: $\displaystyle 2, 6, 18, 54, \ldots$ becomes $\displaystyle 2, -6, 18, -54, \ldots$ – a geometric sequence with ratio $\displaystyle -q$ .

Exercise 1

The geometric sequence: $\displaystyle 3, 6, 12, \ldots$
Calculate the sum of the first 8 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 geometric sequence: $\displaystyle 1, -2, 4, \ldots$
Calculate $\displaystyle S_9$ .

/ – 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 3

The geometric sequence: $\displaystyle 162, 54, 18, \ldots$
Calculate the sum of the first 5 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 4

A geometric sequence has $\displaystyle {a_1}={5}$ , $\displaystyle {q}={2}$ and 10 terms.
Calculate the sum of the terms in the even places.

/ – 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 5

The sequence: $\displaystyle {b_n}={2\cdot3^{n-1}}$ .

a

Write the sum $\displaystyle {S_n}={2+6+18+\ldots+2\cdot3^{n-1}}$ in terms of $\displaystyle n$ , and simplify.

/ – 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

b

A new sequence is made from $\displaystyle b_n$ : the sign of the terms in the even places is changed.
Find the sum of the first 36 terms of the new sequence.

/ – 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

Formulas: sequences, grade 12 (4 units)

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