Arithmetic Progressions
Yaritza signs up for a training plan to run the Santo Domingo Marathon: on the first day she runs 12 km, and each day she increases the distance by the same amount. On what day will she reach 42 km? How many kilometers will she have run in total? When a fixed amount is always added to each term, you have an arithmetic progression. In this section you will learn to prove its formulas and to use them in problems with equations and systems.
General Term and Common Difference
Why? To get from $a_1$ to $a_n$ you take $n-1$ steps, and each step adds $d$. That is the whole proof: $a_n = a_1 + \underbrace{d + d + \dots + d}_{n-1}$.
To check whether a sequence with a known general term is arithmetic, calculate $a_{n+1}-a_n$: if the result does not depend on $n$, it is the common difference. For example, with $b_n = 5-3n$: $b_{n+1}-b_n = (5-3(n+1))-(5-3n) = -3$. It is constant, so the sequence is arithmetic with $d=-3$.
To find out whether a number belongs to the progression, set $a_n$ equal to that number and solve: if $n$ comes out as a natural number, the number is in that position; if not, it does not belong.
Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$
Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$
Given the sequence defined by the general term $b_n = 5 - 3n$, prove that it is an arithmetic progression and determine its common difference.
Given the sequence defined by the general term $b_n = 5 - 3n$, prove that it is an arithmetic progression and determine its common difference.
Determine whether the number $102$ belongs to the arithmetic progression whose general term is $c_n = 4n - 6$. If so, state its position in the sequence.
Determine whether the number $102$ belongs to the arithmetic progression whose general term is $c_n = 4n - 6$. If so, state its position in the sequence.
The first three terms of an arithmetic progression are $x$, $x+4$ and $2x-1$. Calculate the numerical value of $x$ and write down these first three terms.
The first three terms of an arithmetic progression are $x$, $x+4$ and $2x-1$. Calculate the numerical value of $x$ and write down these first three terms.
Finding Data with Equations and Systems
If you know two non-consecutive terms, you can set up a system with $a_1$ and $d$ as the unknowns, or count the "steps" between them.
The eighth term of an arithmetic progression is $32$ and its common difference is $d = 4$. Find the first term $a_1$ and the expression for its general term $a_n$.
The eighth term of an arithmetic progression is $32$ and its common difference is $d = 4$. Find the first term $a_1$ and the expression for its general term $a_n$.
In an arithmetic progression, it is known that the fourth term is $15$ and the tenth term is $39$. Set up a system of equations to calculate the common difference $d$…
In an arithmetic progression, it is known that the fourth term is $15$ and the tenth term is $39$. Set up a system of equations to calculate the common difference $d$ and the first term $a_1$. Subsequently, find the sum of the first $20$ terms of this progression.
Interpolate five arithmetic means between the numbers $-8$ and $22$. Write down the complete resulting arithmetic progression, stating its common difference.
Interpolate five arithmetic means between the numbers $-8$ and $22$. Write down the complete resulting arithmetic progression, stating its common difference.
Determine the values of the parameter $k$ such that the algebraic expressions $k^2 - 2$, $3k + 1$ and $k^2 + 4k$ are three consecutive terms of an arithmetic…
Determine the values of the parameter $k$ such that the algebraic expressions $k^2 - 2$, $3k + 1$ and $k^2 + 4k$ are three consecutive terms of an arithmetic progression. For each value of $k$ found, write down the corresponding three terms.
The Sum of the Terms
Legend has it that the young Gauss added 1 through 100 in seconds: he paired $1+100$, $2+99$, $3+98$… and each pair adds up to $101$. There are $50$ pairs, so the total is $5{,}050$.
Proof. Write the sum twice, once in reverse:
$S_n = a_1 + a_2 + \dots + a_n$
$S_n = a_n + a_{n-1} + \dots + a_1$
Adding column by column, each pair equals $a_1+a_n$ (because when you move forward one term in one row and backward in the other, what is added and subtracted is $d$). There are $n$ pairs: $2S_n = n(a_1+a_n)$.
Another form that works is $S_n = \dfrac{n\,[2a_1+(n-1)d]}{2}$, which is obtained by substituting $a_n$.
To add the terms from position $p$ through position $q$ (inclusive), you can subtract: $S_q - S_{p-1}$. Or treat them as a progression of $q-p+1$ terms, with first term $a_p$ and last term $a_q$.
Calculate the sum of the first $30$ terms of an arithmetic progression, knowing that the first term is $a_1 = 4$ and the thirtieth term is $a_{30} = 91$.
Calculate the sum of the first $30$ terms of an arithmetic progression, knowing that the first term is $a_1 = 4$ and the thirtieth term is $a_{30} = 91$.
Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).
Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).
In an arithmetic progression, the first term is $a_1 = 3$ and the common difference is $d = 2$. If the sum of the first $n$ terms is $S_n = 360$, set up the…
In an arithmetic progression, the first term is $a_1 = 3$ and the common difference is $d = 2$. If the sum of the first $n$ terms is $S_n = 360$, set up the corresponding quadratic equation and determine the number of terms $n$ that have been added.
In an arithmetic progression with common difference $d = 3$, the first term is $a_1 = -10$. Calculate the sum of the terms between position $20$ and position $40$,…
In an arithmetic progression with common difference $d = 3$, the first term is $a_1 = -10$. Calculate the sum of the terms between position $20$ and position $40$, inclusive.
Problems and Loans
Always ask yourself: Is the same amount added each time? Am I being asked for a term or a sum? How many terms are there?
Simple interest
If you deposit a principal $C$ in a credit union at an annual simple interest rate $i$, the same interest $Ci$ is paid each year on the original principal. The accumulated amount is an arithmetic progression: $a_t = C + t\cdot Ci$, with common difference $d = Ci$.
Loans with increasing payments
A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $3\,000$ pesos, in the second month…
A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $3\,000$ pesos, in the second month $3\,900$ pesos, in the third month $4\,800$ pesos, and so on. How much money in total will the employee have received in bonuses after $2$ full years of work?
A cyclist begins her training plan by riding $12$ km on the first day. To improve her endurance, each day she increases the distance of her ride by $1.5$ km compared to…
A cyclist begins her training plan by riding $12$ km on the first day. To improve her endurance, each day she increases the distance of her ride by $1.5$ km compared to the previous day. On which day of her training will she ride exactly $42$ km? How many total kilometers will she have ridden, adding up all the days up to and including that day?
A theater has its seats arranged in a fan shape. The first row has $20$ seats, the second $24$, the third $28$, and so on in arithmetic progression. If the theater has a…
A theater has its seats arranged in a fan shape. The first row has $20$ seats, the second $24$, the third $28$, and so on in arithmetic progression. If the theater has a total of $15$ rows, how many seats are in the last row? What is the total capacity (total number of seats) of the theater?
A freely falling object on a fictional planet travels $4.9$ m in the first second. In each subsequent second, the distance travelled during that second increases by…
A freely falling object on a fictional planet travels $4.9$ m in the first second. In each subsequent second, the distance travelled during that second increases by $9.8$ m compared to the distance travelled in the previous second. What distance does the object travel exclusively during the tenth second of its fall?
To reforest a wood after a fire, a forestry team plants trees progressively. In the first week, they manage to plant $120$ trees, but due to accumulated fatigue and less…
To reforest a wood after a fire, a forestry team plants trees progressively. In the first week, they manage to plant $120$ trees, but due to accumulated fatigue and less available accessible land, each week they plant $5$ fewer trees than the previous week. How many weeks will pass until the team has planted an exact total of $1470$ trees? (Note: Analyse the possible solutions mathematically and justify which one makes sense in the context of the problem).
The interior angles of a convex hexagon are in arithmetic progression. Knowing that the largest angle measures $140^\circ$, calculate the measures of the other five…
The interior angles of a convex hexagon are in arithmetic progression. Knowing that the largest angle measures $140^\circ$, calculate the measures of the other five interior angles of the hexagon. (Remember that the sum of the interior angles of an $n$-sided polygon is $180^\circ \times (n-2)$).
In an industrial warehouse, cylindrical steel pipes are stacked such that each row has exactly one pipe fewer than the row immediately below it on which it rests. If the…
In an industrial warehouse, cylindrical steel pipes are stacked such that each row has exactly one pipe fewer than the row immediately below it on which it rests. If the top row (the peak) has just $1$ pipe and there are $210$ stacked pipes in total, how many rows of pipes make up the stack? How many pipes are in the bottom row resting on the ground?
A company purchases an industrial machine for $1\,920\,000$ pesos and reaches an agreement to pay for it in monthly instalments that form an arithmetic progression. If…
A company purchases an industrial machine for $1\,920\,000$ pesos and reaches an agreement to pay for it in monthly instalments that form an arithmetic progression. If it is known that the first instalment to be paid is $30\,000$ pesos and the last instalment will be $162\,000$ pesos, in how many months will the company settle its debt? What is the exact monthly increase in the instalment compared to the previous month?
Proofs with Progressions
Some exercises ask you to prove something. Two ideas are enough:
For three numbers in arithmetic progression, it helps to write them as $x-d,\ x,\ x+d$: that way their sum is $3x$ and things simplify a lot. Also, $a,b,c$ are in arithmetic progression exactly when $2b=a+c$.
The sum of the first $n$ terms of a sequence is given by the expression $S_n = 2n^2 + 5n$. Find the mathematical expression for the general term $a_n$ and prove…
The sum of the first $n$ terms of a sequence is given by the expression $S_n = 2n^2 + 5n$. Find the mathematical expression for the general term $a_n$ and prove rigorously that this sequence is an arithmetic progression.
Find three numbers in arithmetic progression knowing that their sum is $24$ and that the sum of their squares is $242$.
Find three numbers in arithmetic progression knowing that their sum is $24$ and that the sum of their squares is $242$.
Prove algebraically that if the numbers $a^2$, $b^2$ and $c^2$ are in arithmetic progression, then the numbers $\frac{1}{b+c}$, $\frac{1}{a+c}$ and $\frac{1}{a+b}$ also…
Prove algebraically that if the numbers $a^2$, $b^2$ and $c^2$ are in arithmetic progression, then the numbers $\frac{1}{b+c}$, $\frac{1}{a+c}$ and $\frac{1}{a+b}$ also form an arithmetic progression.
Back to the beginning: Yaritza runs $a_n = 12+(n-1)d$ km; with the $d$ given in her plan, you solve $a_n=42$ for the day and use $S_n$ for the total.