Geometrical Progression

Excerpts

Geometrical Progression
A series of numbers is said to be in **Geometrical Progression** when the quotient of any term divided by the preceding term is the same throughout the series.
Geometrical Progression
Hence the $n$th term will be $ar^{n - 1}$.
Geometrical Progression
In formula (1), put $5$ for $n$, $3$ for $a$, and $2$ for $r$.
Geometrical Progression
the geometrical mean of any two numbers is the square root of their product.
Geometrical Progression
Therefore the series is $3$, $±6$, $12$, $±24$, $\dots$.
Geometrical Progression
If a blacksmith uses seven nails in putting a shoe on a horse’s foot, and receives $1$ cent for the first nail, $2$ cents for the second nail, and so on, what does he receive for putting on the shoe?

Equations

Geometrical Progression
\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}
Successive terms a, b, c, d are in geometrical progression when each term divided by the one before it gives the same quotient.
Geometrical Progression
l = ar^{n - 1}
The nth term l of a geometrical progression equals the first term a times the common ratio r raised to the power n minus 1.
Geometrical Progression
G = ± \sqrt{ab}
The geometrical mean G of two numbers a and b is plus or minus the square root of their product, which is equivalent to G squared equal to ab.
Geometrical Progression
s = \frac{a(r^{n} - 1)}{r - 1}
The sum s of the first n terms of a geometrical progression equals a times (r to the power n minus 1) divided by (r minus 1).
Geometrical Progression
s = \frac{a(1 - r^{n})}{1 - r}
For a common ratio r less than 1, the sum of n terms of a geometrical progression can be written as a times (1 minus r to the power n) divided by (1 minus r).

Problems

Exercise 77

Exercise 77, problem 1, p. 151
Find the 5th term of $3$, $9$, $27$ $\dots$.
Printed answer:
- $243$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 243
On the STU-32 (STU, rpn):
```
3 ENTER 5 ENTER 1 − yˣ
```
```
3 ×
```
Calculator: +243E+0; the book prints 243. Run on the calculator core at firmware 628c96c.
Exercise 77, problem 10, p. 151
$8$, $4$, $2$, $\dots$ to $8$ terms.
Printed answer:
- $15\frac{15}{16}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(255, 16)
Exercise 77, problem 11, p. 151
$64$, $32$, $16$, $\dots$ to $9$ terms.
Printed answer:
- $127\frac{3}{4}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(511, 4)
Exercise 77, problem 12, p. 151
$64$, $-32$, $16$, $\dots$ to $5$ terms.
Printed answer:
- $44$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 44
Exercise 77, problem 13, p. 151
$\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{9}$, $\dots$ to $4$ terms.
Printed answer:
- $1\frac{11}{54}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(65, 54)
Exercise 77, problem 14, p. 151
If a blacksmith uses seven nails in putting a shoe on a horse’s foot, and receives $1$ cent for the first nail, $2$ cents for the second nail, and so on, what does he receive for putting on the shoe?
Printed answer:
- $$1.27$.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: FLAG-PARSE Rational(127, 100)
Exercise 77, problem 15, p. 151
If a boy receives $2$ cents for his first day’s work, $4$ cents for his second day, $8$ cents for the third day, and so on for $12$ days, what will his wages amount to?
Printed answer:
- $$81.90$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(8190, 100)
Exercise 77, problem 16, p. 151
If the population of a city is $10,000$, and increases $10$% a year for four years, what will be its population at the end of the four years? (Here $l = ar^{4}$.)
Printed answer:
- $14,641$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 14641
Exercise 77, problem 2, p. 151
Find the 7th term of $3$, $6$, $12$ $\dots$.
Printed answer:
- $192$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 192
On the STU-32 (STU, rpn):
```
6 ENTER 3 ÷ ENTER 7 ENTER 1 − yˣ 3 ×
```
Calculator: +192E+0; the book prints 192. Run on the calculator core at firmware 628c96c.
Exercise 77, problem 3, p. 151
Find the 8th term of $6$, $3$, $\frac{3}{2}$ $\dots$.
Printed answer:
- $\frac{3}{64}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(3, 64)
Exercise 77, problem 4, p. 151
Find the 9th term of $1$, $-2$, $4$ $\dots$.
Printed answer:
- $256$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 256
On the STU-32 (STU, rpn):
```
2 +/− ENTER 9 ENTER 1 − yˣ
```
Calculator: +256E+0; the book prints 256. Run on the calculator core at firmware 628c96c.
Exercise 77, problem 5, p. 151
Find the geometrical mean between $2$ and $8$.
Printed answer:
- $±4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-4, 4]
Exercise 77, problem 6, p. 151
Find the common ratio if the 1st and 3d terms are $2$ and $32$.
Printed answer:
- $±4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-4, 4]
Exercise 77, problem 7, p. 151
$3$, $9$, $27$, $\dots$ to $6$ terms.
Printed answer:
- $1092$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 1092
On the STU-32 (STU, rpn):
```
3 ENTER 6 yˣ 1 −
```
```
3 ×
```
```
3 ENTER 1 −
```
```
÷
```
Calculator: +1092E+0; the book prints 1092. Run on the calculator core at firmware 628c96c.
Exercise 77, problem 8, p. 151
$3$, $6$, $12$, $\dots$ to $8$ terms.
Printed answer:
- $765$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 765
On the STU-32 (STU, rpn):
```
6 ENTER 3 ÷
```
```
ENTER ENTER 8 yˣ 1 −
```
```
x↔y 1 − ÷
```
```
3 ×
```
Calculator: +765E+0; the book prints 765. Run on the calculator core at firmware 628c96c.
Exercise 77, problem 9, p. 151
$6$, $3$, $\frac{3}{2}$, $\dots$ to $7$ terms.
Printed answer:
- $11\frac{29}{32}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(381, 32)