Arithmetical Progression

Excerpts

Arithmetical Progression
A series of numbers is said to form an **Arithmetical Progression** if the difference between any term and the preceding term is the same throughout the series.
Arithmetical Progression
the coefficient of $d$ in each term being always less by $1$ than the *number of the term*.
Arithmetical Progression
it is evident that the series beginning with the first term will be $a$, $a + d$, $a + 2d$, etc., and beginning with the last term will be $l$, $l - d$, $l - 2d$, etc.
Arithmetical Progression
Hence, the arithmetical mean of any two numbers is found by taking half their sum.
Arithmetical Progression
Putting for $l$ its value $a + (n - 1)d$, in formula (4), we have
Arithmetical Progression
If $d$ is positive, the progression is an *increasing* series; if $d$ is negative, the progression is a *decreasing* series.
Arithmetical Progression
A heavy body falling from a height falls $16.1$ feet the first second, and in each succeeding second $32.2$ feet more than in the second next preceding.

Equations

Arithmetical Progression
l = a + (n - 1)d
The nth term l of an arithmetical progression equals the first term a plus (n - 1) times the common difference d.
Arithmetical Progression
A = \frac{a + b}{2}
If A is the arithmetical mean of a and b, then A is half their sum.
Arithmetical Progression
m + 2 = n
The total number of terms n is the number of inserted means m plus the two given end numbers.
Arithmetical Progression
l = a + (m + 1)d
The last term l equals the first term a plus (m + 1) times the common difference, when m means are inserted.
Arithmetical Progression
\frac{l - a}{m + 1} = d
The common difference when m arithmetical means are inserted between a and l is the difference l - a divided by m + 1.
Arithmetical Progression
2s = n(a + l)
Twice the sum of the terms equals the number of terms times the sum of the first and last terms.
Arithmetical Progression
s = \frac{n}{2}(a + l)
The sum of n terms of an arithmetical progression is n over 2 times the sum of the first and last terms (Formula (4)).
Arithmetical Progression
s = \frac{n}{2}\bigl\{a + a + (n - 1)d\bigr\}
The sum of n terms of an arithmetical progression, with l replaced by a + (n - 1)d, is n over 2 times the braced sum of a, a and (n - 1)d (Formula (5), first line).

Problems

Exercise 75

Exercise 75, problem 1, p. 144
Find the 25th term in the series $3$, $6$, $9$, $\dots$.
Printed answer:
- $75$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 75
Exercise 75, problem 10, p. 144
What is the arithmetical mean of $20$ and $32$?
Printed answer:
- $26$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 26
Exercise 75, problem 11, p. 144
What is the arithmetical mean of $a + b$ and $a - b$?
Printed answer:
- $a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a
Exercise 75, problem 12, p. 144
Insert $8$ arithmetical means between $20$ and $29$.
Printed answer:
- $21$, $22$, etc.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 75, problem 2, p. 144
Find the 13th term in the series $50$, $49$, $48$, $\dots$.
Printed answer:
- $38$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 38
Exercise 75, problem 3, p. 144
Find the 15th term in the series $\frac{1}{7}$, $\frac{3}{7}$, $\frac{5}{7}$, $\dots$.
Printed answer:
- $4\frac{1}{7}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(29, 7)
Exercise 75, problem 4, p. 144
Find the 19th term in the series $\frac{1}{4}$, $-\frac{1}{4}$, $-\frac{3}{4}$, $\dots$.
Printed answer:
- $-8\frac{3}{4}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(-35, 4)
Exercise 75, problem 5, p. 144
Find the 10th term in an arithmetical progression whose 1st term is $5$ and 3d term $9$.
Printed answer:
- $23$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"T": 23}
Exercise 75, problem 6, p. 144
Find the 11th term in an arithmetical progression whose 1st term is $10$ and whose 6th term is $5$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"T": 0}
Exercise 75, problem 7, p. 144
If the 3d term of an arithmetical progression is $20$ and the 13th term is $100$, what is the 20th term?
Printed answer:
- $156$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"T": 156}
Exercise 75, problem 8, p. 144
Which term of the series $5$, $7$, $9$, $11$, $\dots$, is $43$?
Printed answer:
- $20$th.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 20
Exercise 75, problem 9, p. 144
Which term of the series $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\dots$, is $18$?
Printed answer:
- $101$st.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 101

Exercise 76

Exercise 76, problem 1, p. 146
Find the sum of $3$, $5$, $7$, $\dots$, to $20$ terms.
Printed answer:
- $440$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 440
On the STU-32 (STU, rpn):
```
2 ENTER 3 ×
```
```
20 ENTER 1 −
```
```
2 ×
```
```
+
```
```
20 ENTER 2 ÷
```
```
×
```
Calculator: +440E+0; the book prints 440. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 10, p. 146
In a potato race each man picked up $50$ potatoes placed in line a yard apart, and the first potato one yard from the basket, picking up one potato at a time and bringing it to the basket. How many yards did each man run, the start being made from the basket?
Printed answer:
- $2550$ yd.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 2550
On the STU-32 (STU, rpn):
```
50 ENTER 1 +
```
```
50 ×
```
Calculator: +2550E+0; the book prints 2550. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 11, p. 146
A heavy body falling from a height falls $16.1$ feet the first second, and in each succeeding second $32.2$ feet more than in the second next preceding. How far will a body fall in $19$ seconds?
Printed answer:
- $5812.1$ ft.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(58121, 10)
On the STU-32 (STU, rpn):
```
16.1 ENTER 2 ×
```
```
18 ENTER 32.2 ×
```
```
+
```
```
19 ENTER 2 ÷
```
```
×
```
Calculator: +581210E-2; the book prints 5812.1. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 12, p. 146
A stone dropped from a bridge reached the water in just $3$ seconds. Find the height of the bridge. (See Ex. 11.)
Printed answer:
- $144.9$ ft.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(1449, 10)
On the STU-32 (STU, rpn):
```
3 ENTER 2 ÷
```
```
16.1 ENTER 2 × 32.2 ENTER 2 × + ×
```
Calculator: +14490E-2; the book prints 144.9. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 13, p. 146
The arithmetical mean between two numbers is $13$, and the mean between the double of the first and the triple of the second is $33\frac{1}{2}$. Find the numbers.
Printed answer:
- $11$, $15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {a: 11, b: 15}
Exercise 76, problem 14, p. 146
Find three numbers of an arithmetical series whose sum shall be $27$, and the sum of the first and second shall be $frac{4}{5}$ of the sum of the second and third.
Printed answer:
- $7$, $9$, $11$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 9, y: 2}
Exercise 76, problem 15, p. 146
A travels uniformly $20$ miles a day; B travels $8$ miles the first day, $12$ the second, and so on, in arithmetical progression. If they start Monday morning from the same place and travel in the same direction, how far apart will they be Saturday night?
Printed answer:
- $12$ miles.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 12
On the STU-32 (STU, rpn):
```
20 ENTER 6 ×
```
```
12 ENTER 8 −
```
```
6 ENTER 1 − ×
```
```
8 ENTER 2 × +
```
```
6 ENTER 2 ÷ ×
```
```
−
```
Calculator: +12E+0; the book prints 12. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 16, p. 146
The sum of three terms of an arithmetical progression is $36$, and the square of the mean exceeds the product of the other two terms by $49$. Find the numbers.
Printed answer:
- $5$, $12$, $19$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 12, y: 7}
Exercise 76, problem 2, p. 146
Find the sum of $14$, $14\frac{1}{2}$, $15$, $\dots$, to $12$ terms.
Printed answer:
- $201$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 201
On the STU-32 (STU, rpn):
```
12 ENTER 1 − 2 ÷
```
```
14 ENTER +
```
```
+
```
```
12 × 2 ÷
```
Calculator: +2010E-1; the book prints 201. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 3, p. 146
Find the sum of $\frac{7}{6}$, $1$, $\frac{5}{6}$, $\dots$, to $10$ terms.
Printed answer:
- $4frac{1}{6}$.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: FLAG-PARSE Rational(25, 6)
Exercise 76, problem 4, p. 146
Find the sum of $-7$, $-5$, $-3$, $\dots$, to $16$ terms.
Printed answer:
- $128$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 128
On the STU-32 (STU, rpn):
```
2 ENTER 7 +/− ×
```
```
16 ENTER 1 − 2 × +
```
```
16 ENTER 2 ÷ ×
```
Calculator: +128E+0; the book prints 128. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 5, p. 146
Find the sum of $12$, $9$, $6$, $\dots$, to $21$ terms.
Printed answer:
- $-378$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes -378
On the STU-32 (STU, rpn):
```
9 ENTER 12 −
```
```
21 ENTER 1 −
```
```
×
```
```
12 ENTER 2 × +
```
```
21 × 2 ÷
```
Calculator: -378E+0; the book prints -378. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 6, p. 146
Find the sum of $-10\frac{1}{2}$, $-9$, $-7\frac{1}{2}$, $\dots$, to $25$ terms.
Printed answer:
- $187\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(375, 2)
Exercise 76, problem 7, p. 146
The sum of three numbers in arithmetical progression is $9$, and the sum of their squares is $35$. Find the numbers.
Printed answer:
- $1$, $3$, $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 3, y: 2}
Exercise 76, problem 8, p. 146
A common clock strikes the hours from $1$ to $12$. How many times does it strike every $24$ hours?
Printed answer:
- $156$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 156
On the STU-32 (STU, rpn):
```
12 ENTER 2 ÷
```
```
12 ENTER 1 + ×
```
```
2 ×
```
Calculator: +156E+0; the book prints 156. Run on the calculator core at firmware 628c96c.
Exercise 76, problem 9, p. 146
The Greenwich clock strikes the hours from $1$ to $24$. How many times does it strike in $24$ hours?
Printed answer:
- $300$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 300
On the STU-32 (STU, rpn):
```
1 ENTER 24 +
```
```
24 ENTER 2 ÷
```
```
×
```
Calculator: +300E+0; the book prints 300. Run on the calculator core at firmware 628c96c.