Arithmetic and geometric sequences together

Sequences, grade 12 (4 units) Lesson 19 of 20Learning

Arithmetic and geometric sequences together

Example: from an arithmetic to a geometric sequence

Three numbers form an arithmetic sequence, and their sum is 15.
Adding 1 to the first number, 1 to the second and 4 to the third gives a geometric sequence.
What are the numbers?

The solution below writes the numbers around the middle one, and uses the geometric mean:

The numbers: $\displaystyle 5-d$ , $\displaystyle 5$ , $\displaystyle 5+d$ – the middle one is the mean, $\displaystyle {\frac{15}{3}}={5}$ .
After the additions: $\displaystyle 6-d$ , $\displaystyle 6$ , $\displaystyle 9+d$ , so $\displaystyle {36}={(6-d)(9+d)}$ .
$\displaystyle {d^2+3d-18}={0}$ , so $\displaystyle {d}={3}$ or $\displaystyle {d}={-6}$ .
$\displaystyle {d}={3}$ : the numbers $\displaystyle 2, 5, 8$ , and the geometric sequence $\displaystyle 3, 6, 12$ .
$\displaystyle {d}={-6}$ : the numbers $\displaystyle 11, 5, -1$ , and the geometric sequence $\displaystyle 12, 6, 3$ .

Example: a run there and back

Dana leaves her house for a run.
In the first minute she covers 80 metres, and in each further minute 1.02 times the distance of the minute before.
After 20 minutes she turns round and comes back the same way.
On the way back she covers 100 metres in the first minute, and in each further minute 5 metres more than in the minute before.
Will she be home within 10 minutes?

Out – a geometric sequence: $\displaystyle {S_{20}}={\frac{80(1.02^{20}-1)}{1.02-1}\approx1943.8}$ metres.
Back – an arithmetic sequence: in 10 minutes $\displaystyle {S_{10}}={\frac{10(2\cdot100+9\cdot5)}{2}}={1225}$ metres.
$\displaystyle 1225<1943.8$ : Dana will not be home within 10 minutes.

Exercise 1

Two savings plans, both paying in 1,000 ₪ in the first month:

In plan A each month’s payment is 100 ₪ more than the month before.
In plan B each month’s payment is 1.05 times the month before.

a

How much is paid into plan A in 12 months?

/ – 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

How much is paid into plan B in 12 months? Round to whole shekels.

/ – 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

c

In which plan is more paid in over 48 months? Work it out and explain.

Plan A

/ – 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

Plan B

/ – 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

In an arithmetic sequence $\displaystyle {a_1}={2}$ and $\displaystyle d\neq0$ .
Its first, second and fourth terms (in this order) form a geometric sequence.
Find $\displaystyle d$ .

\displaystyle d

/ – 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: $\displaystyle 3, 6, 12, \ldots$
The first term of an arithmetic sequence equals the first term of the geometric one, and its third term equals the geometric one’s third term.
Calculate the tenth term of the arithmetic 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

Exercise 4

Three numbers form an arithmetic sequence, and their sum is 24.
Adding 2 to the first number and 2 to the third gives a geometric sequence.
Find the numbers.

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